Topologically guaranteed univariate solutions of underconstrained polynomial systems via no-loop and single-component tests

We present an algorithm which robustly computes the intersection curve(s) of an underconstrained piecewise polynomial system consisting of n equations with n+1 unknowns. The solution of such a system is typically a curve in Rⁿ⁺¹. This work extends the single solution test of Hanniel and Elber (2007) [6] for a set of algebraic constraints from zero-dimensional solutions to univariate solutions, in Rⁿ⁺¹. Our method exploits two tests: a no-loop test (NLT) and a single-component test (SCT) that together isolate and separate domains D where the solution curve consists of just one single component. For such domains, a numerical curve tracing is applied. If one of those tests fails, D is subdivided. Finally, the single components are merged together and, consequently, the topological configuration of the resulting curve is guaranteed. Several possible applications of the solver, namely solving the surface–surface intersection problem, computing 3D trisector curves, flecnodal curves or kinematic simulations in 3D are also discussed.     

Hilbert曲线的快速生成算法设计与实现

研究了Hilbert曲线的特征和现有经典算法，依据二分技术提出了一种全新的空间填充曲线生成算法，算法按照复制的思想将具有“形”特征的曲线问题转化为具有“数”特征的矩阵问题．因此对曲线的操作就转化为对矩阵的运算，而矩阵运算不用考虑绘制曲线方向问题，也不用考虑曲线始点和终点．实验结果表明，该算法比经典的L系统算法提高了将近1倍的速度，有意义的是，该算法为并行计算大型空间填充曲线提出了一种方案。     

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。     

Processor-time optimal parallel algorithms for digitized images on mesh-connected processor arrays

We present processor-time optimal parallel algorithms for several problems onn ×n digitized image arrays, on a mesh-connected array havingp processors and a memory of sizeO(n ²) words. The number of processorsp can vary over the range [1,n ^3/2] while providing optimal speedup for these problems. The class of image problems considered here includes labeling the connected components of an image; computing the convex hull, the diameter, and a smallest enclosing box of each component; and computing all closest neighbors. Such problems arise in medium-level vision and require global operations on image pixels. To achieve optimal performance, several efficient data-movement and reduction techniques are developed for the proposed organization.This research was supported in part by the National Science Foundation under Grant IRI-8710836 and in part by DARPA under Contract F33615-87-C-1436 monitored by the Wright Patterson Airforce Base.     

The μ-basis of a planar rational curve—properties and computation

A moving line L(x,y;t)=0 is a family of lines with one parameter t in a plane. A moving line L(x,y;t)=0 is said to follow a rational curve P(t) if the point P(t₀) is on the line L(x,y;t₀)=0 for any parameter value t₀. A μ-basis of a rational curve P(t) is a pair of lowest degree moving lines that constitute a basis of the module formed by all the moving lines following P(t), which is the syzygy module of P(t). The study of moving lines, especially the μ-basis, has recently led to an efficient method, called the moving line method, for computing the implicit equation of a rational curve [3 and 6]. In this paper, we present properties and equivalent definitions of a μ-basis of a planar rational curve. Several of these properties and definitions are new, and they help to clarify an earlier definition of the μ-basis [3]. Furthermore, based on some of these newly established properties, an efficient algorithm is presented to compute a μ-basis of a planar rational curve. This algorithm applies vector elimination to the moving line module of P(t), and has O(n²) time complexity, where n is the degree of P(t). We show that the new algorithm is more efficient than the fastest previous algorithm [7].     

Interpolating an Unlimited Number of Curves Meeting at Extraordinary Points on Subdivision Surfaces*

Interpolating curves by subdivision surfaces is one of the major constraints that is partially addressed in the literature. So far, no more than two intersecting curves can be interpolated by a subdivision surface such as Doo‐Sabin or Catmull‐Clark surfaces. One approach that has been used in both of theses surfaces is the polygonal complex approach where a curve can be defined by a control mesh rather than a control polygon. Such a definition allows a curve to carry with it cross derivative information which can be naturally embodied in the mesh of a subdivision surface. This paper extends the use of this approach to interpolate an unlimited number of curves meeting at an extraordinary point on a subdivision surface. At that point, the curves can all meet with either C ⁰ or C ¹ continuity, yet still have common tangent plane. A straight forward application is the generation of subdivision surfaces through 3‐regular meshes of curves for which an easy interface can be used.     

Ellipse-based principal component analysis for self-intersecting curve reconstruction from noisy point sets

Surface reconstruction from cross cuts usually requires curve reconstruction from planar noisy point samples. The output curves must form a possibly disconnected 1-manifold for the surface reconstruction to proceed. This article describes an implemented algorithm for the reconstruction of planar curves (1-manifolds) out of noisy point samples of a self-intersecting or nearly self-intersecting planar curve C. C:[a,b]⊂R→R ² is self-intersecting if C(u)=C(v), u≠v, u,v∈(a,b) (C(u) is the self-intersection point). We consider only transversal self-intersections, i.e. those for which the tangents of the intersecting branches at the intersection point do not coincide (C′(u)≠C′(v)). In the presence of noise, curves which self-intersect cannot be distinguished from curves which nearly self-intersect. Existing algorithms for curve reconstruction out of either noisy point samples or pixel data, do not produce a (possibly disconnected) Piecewise Linear 1-manifold approaching the whole point sample. The algorithm implemented in this work uses Principal Component Analysis (PCA) with elliptic support regions near the self-intersections. The algorithm was successful in recovering contours out of noisy slice samples of a surface, for the Hand, Pelvis and Skull data sets. As a test for the correctness of the obtained curves in the slice levels, they were input into an algorithm of surface reconstruction, leading to a reconstructed surface which reproduces the topological and geometrical properties of the original object. The algorithm robustly reacts not only to statistical non-correlation at the self-intersections (non-manifold neighborhoods) but also to occasional high noise at the non-self-intersecting (1-manifold) neighborhoods.     

Curvature continuous offset approximation based on circle approximation using quadratic Bézier biarcs

We present a method for G² end-point interpolation of offset curves using rational Bézier curves. The method is based on a G² end-point interpolation of circular arcs using quadratic Bézier biarcs. We also prove the invariance of the Hausdorff distance between two compatible curves under convolution. Using this result, we obtain the exact Hausdorff distance between an offset curve and its approximation by our method. We present the approximation algorithm and give numerical examples.     

Reparameterization of piecewise rational Bezier curves and its applications

degree . Although the curve segments are C ¹ continuous in three dimensions, they may be C ⁰ continuous in four dimensions. In this case, the multiplicity of each interior knot cannot be reduced and the B-spline basis function becomes C ⁰ continuous. Using a surface generation method, such as skinning these kinds of rational B-spline curves to construct an interpolatory surface, may generate surfaces with C ⁰ continuity. This paper presents a reparameterization method for reducing the multiplicity of each interior knot to make the curve segments C ¹ continuous in four dimensions. The reparameterized rational B-spline curve has the same shape and degree as before and also has a standard form. Some applications in skinned surface and ruled surface generation based on the reparameterized curves are shown. Published online: 19 July 2001     

Equivalence of distinct characterizations for rational rotation-minimizing frames on quintic space curves

A rotation-minimizing frame on a space curve r(t) is an orthonormal basis (f₁,f₂,f₃) for R³, where f₁=r^′/|r^′| is the curve tangent, and the normal-plane vectors f₂,f₃ exhibit no instantaneous rotation about f₁. Such frames are useful in spatial path planning, swept surface design, computer animation, robotics, and related applications. The simplest curves that have rational rotation-minimizing frames (RRMF curves) comprise a subset of the quintic Pythagorean-hodograph (PH) curves, and two quite different characterizations of them are currently known: (a) through constraints on the PH curve coefficients; and (b) through a certain polynomial divisibility condition. Although (a) is better suited to the formulation of constructive algorithms, (b) has the advantage of remaining valid for curves of any degree. A proof of the equivalence of these two different criteria is presented for PH quintics, together with comments on the generalization to higher-order curves. Although (a) and (b) are both sufficient and necessary criteria for a PH quintic to be an RRMF curve, the (non-obvious) proof presented here helps to clarify the subtle relationships between them.     

The Geometry and Matching of Lines and Curves Over Multiple Views

This paper describes the geometry of imaged curves in two and three views. Multi-view relationships are developed for lines, conics and non-algebraic curves. The new relationships focus on determining the plane of the curve in a projective reconstruction, and in particular using the homography induced by this plane for transfer from one image to another. It is shown that given the fundamental matrix between two views, and images of the curve in each view, then the plane of a conic may be determined up to a two fold ambiguity, but local curvature of a curve uniquely determines the plane. It is then shown that given the trifocal tensor between three views, this plane defines a homography map which may be used to transfer a conic or the curvature from two views to a third. Simple expressions are developed for the plane and homography in each case.A set of algorithms are then described for automatically matching individual line segments and curves between images. The algorithms use both photometric information and the multiple view geometric relationships. For image pairs the homography facilitates the computation of a neighbourhood cross-correlation based matching score for putative line/curve correspondences. For image triplets cross-correlation matching scores are used in conjunction with line/curve transfer based on the trifocal geometry to disambiguate matches. Algorithms are developed for both short and wide baselines. The algorithms are robust to deficiencies in the segment extraction and partial occlusion.Experimental results are given for image pairs and triplets, for varying motions between views, and for different scene types. The methods are applicable to line/curve matching in stereo and trinocular rigs, and as a starting point for line/curve matching through monocular image sequences.     

Curve interpolation based on the canonical arc length parametrization

We use the canonical equations (CE) of differential geometry, a local Taylor series representation of any smooth curve with parameter the arc length, as a unifying framework for the development of new CNC algorithms, capable of interpolating 2D and 3D curves, represented parametrically, implicitly or as surface intersections, with accurate feedrate control. We use a truncated form of the CE to compute a preliminary point, at an arc distance from the last interpolation point selected to achieve a desired feedrate profile. The next interpolation point is derived by projecting the preliminary point on the curve. The coefficients in the CE involve the curve’s curvature, torsion and their arc length derivatives. We provide computing procedures for them for common Cartesian representations, demonstrating the generality of the proposed method. In addition, our algorithms admit corrections, which render them more accurate in terms of the programmed feedrate, compared to existing parametric algorithms of the same order.     

Multi-dimensional dynamic facility location and fast computation at query points

We present O(nlogn) time algorithms for the minimax rectilinear facility location problem in R¹ and R². The algorithms enable, once they terminate, computing the cost of any given query point in O(logn) time. Based on these algorithms, we develop a preprocessing procedure which enables solving the following two problems: Fast computation of the cost of any query point in Rd, and fast solution for the dynamic location problem in R² (namely, in the presence of an additional facility). Finally, we show that the preprocessing always gives a bound on the optimal value, which allows us in many cases to find the optimum fast (for both the traditional and the dynamic location problems in Rd for any d).     

一种基于特征点的矢量地图水印算法

提出一种基于特征点的矢量地图水印算法。在嵌入水印信息前,采用Torus自同构映射对水印图像进行置乱处理,将矢量地图分割为指定数量顶点的曲线集合,选择曲线中角度最小的顶点作为特征点,在地图精度允许的范围内,通过改变该特征点的坐标值,将制作的水印图像重复嵌入到这些特征点中。实验结果表明,该算法不仅能获得较好的地图精度,且对压缩攻击和各种裁剪攻击也具有较强的鲁棒性。     

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。     

The shortest path in a simply-connected domain having a curved boundary

Given two distinct points S and E on a closed parametric curve forming the boundary of a simply-connected domain (without holes), this paper provides an algorithm to find the shortest interior path (SIP) between the two points in the domain. The SIP consists of portions of curves along with straight line segments that are tangential to the curve. The algorithm initially computes point-curve tangents and bitangents using their respective constraints. They are then analyzed further to identify potential tangents. A region check is performed to determine the tangent that will form part of the SIP. Portions of the curve that belong to the SIP are also identified during the process. The SIP is identified without explicitly computing the length of the curves/tangents. The curve is represented using non-uniform rational B-splines (NURBS). Results of the implementation are provided.     

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.     

Restricting Voronoi diagrams to meshes using corner validation

Restricted Voronoi diagrams are a fundamental geometric structure used in many applications such as surface reconstruction from point sets or optimal transport. Given a set of sites V = { v _k}ⁿ_k=1 ? ?^d and a mesh X with vertices in ?^d connected by triangles, the restricted Voronoi diagram partitions X by computing for each site the portion of X for which the site is the nearest. The restricted Voronoi diagram is the intersection between the regular Voronoi diagram and the mesh. Depending on the site distribution or the ambient space dimension computing the regular Voronoi diagram may not be feasible using classical algorithms. In this paper, we extend Lévy and Bonneel's approach [ LB12 ] based on nearest neighbor queries. We show that their method is limited when the sites are not located on X . We propose a new algorithm for computing restricted Voronoi which reduces the number of sites considered for each triangle of the mesh and scales smoothly when the sites are far from the surface.     

Efficiently Approximating Polygonal Paths in Three and Higher Dimensions

We present efficient algorithms for solving polygonal-path approximation problems in three and higher dimensions. Given an n -vertex polygonal curve P in \R ^d , d \geq 3 , we approximate P by another poly- gonal curve P' of m ≤ n vertices in \R ^d such that the vertex sequence of P' is an ordered subsequence of the vertices of P . The goal is either to minimize the size m of P' for a given error tolerance \eps (called the min-\# problem), or to minimize the deviation error \eps between P and P' for a given size m of P' (called the min- \eps problem). Our techniques enable us to develop efficient near-quadratic-time algorithms in three dimensions and subcubic-time algorithms in four dimensions for solving the min-\# and min-\eps problems. We discuss extensions of our solutions to d -dimensional space, where d > 4 , and for the L ₁ and L _∈ fty metrics. Received January 10, 1999; revised November 8, 2000.     

Complete subdivision algorithms, II: Isotopic meshing of singular algebraic curves

Given a real valued function f(X,Y), a box region B₀⊆R² and ε>0, we want to compute an ε-isotopic polygonal approximation to the restriction of the curve S=f⁻¹(0)={p∈R²:f(p)=0} to B₀. We focus on subdivision algorithms because of their adaptive complexity and ease of implementation. Plantinga & Vegter gave a numerical subdivision algorithm that is exact when the curve S is bounded and non-singular. They used a computational model that relied only on function evaluation and interval arithmetic. We generalize their algorithm to any bounded (but possibly non-simply connected) region that does not contain singularities of S. With this generalization as a subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete purely numerical method to compute isotopic approximations of algebraic curves with isolated singularities.