负参数Bézier曲线的性质及其应用

1引言 从便于形状处理、形状信息传递与数据交换的角度来看,形状的数据描述应满足如下要求[1,2]:1.唯一性;2.同一形式性;3.几何不变性;4.易于定界性.在实际应用中往往难以做到的是3、4条.Bezier曲线则利用了Bernstein基函数和等于1来满足几何不变性,再又利用各基函数非负来达到凸包性.易于定界不一定要是凸包的,文[2]中就负权因子分析了这种丧失了凸包性的有理二次Bezier曲线的一些性质,得出用不同的负权因子来生成椭圆弧、抛物线弧和双曲线弧的应用.     

关于环Zn上的椭圆曲线

基于环 上椭圆曲线的射影化定义,深入研究了 上椭圆曲线点群中点的结构形式,讨论了点的加法的射影化定义和仿射化定义,指出在仿射化定义条件下文献[1]和[3]在定义点的加法时存在的一个认识上的错误,给出了仿射化定义条件下两种定义方法之间的区别与联系,最后针对仿射化定义条件下某些点的加法无意义的情形进行了讨论,并分析了导致这种情形的根本原因。     

On Cosmology in Nonlinear Multidimensional Gravity with Multiple Factor Spaces

Within the scope of multidimensional Kaluza–Klein gravity with nonlinear curvature terms and two spherical extra spaces of dimensions m and n, we study the properties of an effective action for the scale factors of extra dimensions. Dimensional reduction leads to an effective 4D multiscalar-tensor theory. Based on qualitative estimates of the Casimir energy contribution at a physically reasonable length scale, we demonstrate the existence of such sets of the initial parameters of the theory in the case m = n that provide a minimum of the effective potential at this scale which yields a fine-tuned value of the effective 4D cosmological constant The corresponding size of extra dimensions depends of which conformal frame is interpreted as the observational one: it is about three orders of magnitude larger than the standard Planck length if we adhere to the Einstein frame, but it is n-dependent in the Jordan frame, and its invisibility requirement restricts the total dimension to values D = 4 + 2n ≤ 20.     

基于小波的曲线任意放大

文章利用小波多分辨理论,提出了任意比例放大曲线的方法.该算法简单,放大后的曲线失真小且比较光滑.同时,在理论上讨论了适于放大缩小曲线的小波基应具有的特征.理论和实验结果均说明双正交B样条小波和半正交B样条小波具有较好的几何特征,适于本文算法.     

On the Detection of Robust Curves

Given m points in the plane and a threshold t, a curve is defined to be robust if at least t points lie on it. Efficient algorithms for detecting robust curves are given; the key contribution is to use randomized sampling. In addition, an approximate version of the problem is introduced. A geometric solution to this problem is given; it too can be enhanced by randomization. These algorithms are readily generalized to solve the problem of robust curve detection in a scene of curve fragments: given a set of curve segments, a curve σ is defined to be robust if curve segments of total length at least l lie on σ. Again, both an exact and an approximate version of the problem are considered. The problems and solutions are closely related to the well-investigated Hough transform technique.     

Shape-From-Template with Curves

International Journal of Computer Vision - Shape-from-Template (SfT) is the problem of using a shape template to infer the shape of a deformable object observed in an image. The usual case of SfT...     

On Shape of Plane Elastic Curves

We study shapes of planar arcs and closed contours modeled on elastic curves obtained by bending, stretching or compressing line segments non-uniformly along their extensions. Shapes are represented as elements of a quotient space of curves obtained by identifying those that differ by shape-preserving transformations. The elastic properties of the curves are encoded in Riemannian metrics on these spaces. Geodesics in shape spaces are used to quantify shape divergence and to develop morphing techniques. The shape spaces and metrics constructed are novel and offer an environment for the study of shape statistics. Elasticity leads to shape correspondences and deformations that are more natural and intuitive than those obtained in several existing models. Applications of shape geodesics to the definition and calculation of mean shapes and to the development of shape clustering techniques are also investigated.     

带形状参数的插值曲线

针对α-B样条插值曲线仅能整体调控其形状的局限性，通过适当地提高曲线次数并引入若干新的形状参数，构造出2种C^2连续的插值曲线，这些曲线的形状既能整体又可局部调控；还给出一种能作整体调控的C^3连续的插值曲线．     

Curves with rational Frenet-Serret motion

This paper examines a special type of rational curves called rational Frenet-Serret (RF) curves distinguished by the property that the motion of their Frenet-Serret frame is rational. It is shown that a rational curve is an RF curve if and only if it has rational speed and rational curvature. The paper derives a general representation formula for RF curves suitable for geometric design and provides a geometric survey of special RF curves. The special case of a cubic helix is examined thoroughly. Additionally the paper discusses several applications including examples for the design of rational sweeping surfaces, rational pipe surfaces and rational transition surfaces joining sweeps with G¹ continuity.     

采用手绘曲线的交互式发型生成方法

将矢量场发型生成方法和手绘曲线交互相结合,提出一种交互式发型生成方法.该方法将发型曲线和发型编辑笔画作为矢量场生成的约束条件,使用户可利用手绘曲线来生成或编辑不同的发型,为发型生成提供了交互手段;同时,采用插值方法对空间矢量场计算进行了简化,确保了交互式发型生成的实时性.实验结果表明:文中方法可满足多种发型交互生成的需要.     

Curves with rational chord-length parametrization

It has been recently proved that rational quadratic circles in standard Bézier form are parameterized by chord-length. If we consider that standard circles coincide with the isoparametric curves in a system of bipolar coordinates, this property comes as a straightforward consequence. General curves with chord-length parametrization are simply the analogue in bipolar coordinates of nonparametric curves. This interpretation furnishes a compact explicit expression for all planar curves with rational chord-length parametrization. In addition to straight lines and circles in standard form, they include remarkable curves, such as the equilateral hyperbola, Lemniscate of Bernoulli and Limaçon of Pascal. The extension to 3D rational curves is also tackled.     

On Convergence of Interpolation Processes in a Hilbert Space

For a Hilbert measure space, estimates of the accuracy of interpolation processes are obtained and theorems on the convergence of these processes to polynomial and integer operators are proved.     

On the Use of Level Curves in Image Analysis

A digitized image is viewed as a surface over the xy-plane. The level curves of this surface provide information about edge directions and feature locations. This paper presents algorithms for the extraction of tangent directions and curvatures of these level curves. The tangent direction is determined by a least-squares minimization over the surface normals (calculated for each 2 × 2 pixel neighborhood) in an averaging window. The curvature calculation, unlike most previous work on this topic, does not require a parameterized curve, but works instead directly on the tangents across adjacent level curves. The curvature is found by fitting concentric circles to the tangent directions via least-squares minimization. The stability of these algorithms with respect to noise is studied via controlled tests on computer generated data corrupted by simulated noise. Examples on real images are given which show application of these algorithms for directional enhancement, and feature point detection.     

On Approximating Polygonal Curves in Two and Three Dimensions

Given a polygonal curve P =[p₁, p₂, . . . , p_n], the polygonal approximation problem considered calls for determining a new curve P′ = [p′₁, p′₂, . . . , p′_m] such that (i) m is significantly smaller than n, (ii) the vertices of P′ are an ordered subset of the vertices of P, and (iii) any line segment [p′_A, p′_{A + 1} of P′ that substitutes a chain [p_B, . . . , p_C] in P is such that for all i where B ≤ i ≤ C, the approximation error of p_i with respect to [p′_A, p′_{A + 1}], according to some specified criterion and metric, is less than a predetermined error tolerance. Using the parallel-strip error criterion, we study the following problems for a curve P in R^d, where d = 2, 3: (i) minimize m for a given error tolerance and (ii) given m, find the curve P′ that has the minimum approximation error over all curves that have at most m vertices. These problems are called the min-# and min-ϵ problems, respectively. For R² and with any one of the L₁, L₂, or L_∞ distance metrics, we give algorithms to solve the min-# problem in O(n²) time and the min-ϵ problem in O(n² log n) time, improving the best known algorithms to date by a factor of log n. When P is a polygonal curve in R³ that is strictly monotone with respect to one of the three axes, we show that if the L₁ and L_∞ metrics are used then the min-# problem can be solved in O(n²) time and the min-ϵ problem can be solved in O(n³) time. If distances are computed using the L₂ metric then the min-# and min-ϵ problems can be solved in O(n³) and O(n³ log n) time, respectively. All of our algorithms exhibit O(n²) space complexity. Finally, we show that if it is not essential to minimize m, simple modifications of our algorithms afford a reduction by a factor of n for both time and space.     

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。     

Visualizing Multidimensional Data with Order Statistics

Multidimensional data sets are common in many domains, and dimensionality reduction methods that determine a lower dimensional embedding are widely used for visualizing such data sets. This paper presents a novel method to project data onto a lower dimensional space by taking into account the order statistics of the individual data points, which are quantified by their depth or centrality in the overall set. Thus, in addition to conveying relative distances in the data, the proposed method also preserves the order statistics, which are often lost or misrepresented by existing visualization methods. The proposed method entails a modification of the optimization objective of conventional multidimensional scaling (MDS) by introducing a term that penalizes discrepancies between centrality structures in the original space and the embedding. We also introduce two strategies for visualizing lower dimensional embeddings of multidimensional data that takes advantage of the coherent representation of centrality provided by the proposed projection method. We demonstrate the effectiveness of our visualization with comparisons on different kinds of multidimensional data, including categorical and multimodal, from a variety of domains such as botany and health care.     

Visualizing Multidimensional Data with Glyph SPLOMs

Scatterplot matrices or SPLOMs provide a feasible method of visualizing and representing multi‐dimensional data especially for a small number of dimensions. For very high dimensional data, we introduce a novel technique to summarize a SPLOM, as a clustered matrix of glyphs, or a Glyph SPLOM. Each glyph visually encodes a general measure of dependency strength, distance correlation, and a logical dependency class based on the occupancy of the scatterplot quadrants. We present the Glyph SPLOM as a general alternative to the traditional correlation based heatmap and the scatterplot matrix in two examples: demography data from the World Health Organization (WHO), and gene expression data from developmental biology. By using both, dependency class and strength, the Glyph SPLOM illustrates high dimensional data in more detail than a heatmap but with more summarization than a SPLOM. More importantly, the summarization capabilities of Glyph SPLOM allow for the assertion of “necessity” causal relationships in the data and the reconstruction of interaction networks in various dynamic systems.     

隐式多项式曲线的信息建模研究进展

用隐式多项式曲线来描述数据点集合轮廓具有天然的优势,尤其是在数据点集合轮廓的拟合过程中体现得更为明显。概括了基于隐式多项式曲线的信息建模研究现状,侧重于目前国内外各种隐式多项式曲线拟合算法的分析以及优劣比较。以多个图像物体数据点集合轮廓为例,使用各种拟合算法对其进行拟合,并给出了拟合的效果,分析了算法的优劣和改进措施以及以后的研究方向.