参数曲面上的插值与混合

如何表示曲面上的曲线,在处理诸如数控加工中的路径设计以及CAD/CAM等领域频繁出现的曲面裁剪问题时显得日益重要.给出了数据点的切方向(切方向及曲率向量或测地曲率值)指定而G¹连续(G²连续)插值曲面上任意点列的方法.作为曲面上曲线插值问题的特例,还讨论了曲面上曲线的混合问题.基本思想是借助于微分几何的有关结论,曲面上曲线的插值问题被转化为其参数平面上类似的曲线插值问题.该方法能够用二维隐式方程来表示曲面上的插值曲线,从而把在显示该曲线时所面对的曲面求交的几何问题转化为计算隐式曲线的代数问题.实验证明该方法是可行的,而且适用于CAD/CAM及计算机图形学等领域.     

一种基于Bezier插值曲面的图像放大方法

章提出了一种利用Ｂｅｚｉｅｒ插值曲面进行图像放大的方法,该方法是为数字图像的第一个色彩分量构造一个分块双三次Ｂｅｚｉｅｒ插值Ｃ＾１曲面的 图像放大等价于以不同的采样速度地该曲面进行采样的过程,实验结果表明,该方法可以大大改善放大图像的效果。     

一类加权有理插值样条曲面及局部约束控制

目的 构造一类新的基于函数值与偏导数值的加权有理插值样条曲面,讨论该样条曲面的相关性质并分析曲面的局部约束控制。方法 一方面,先从x方向构造有理三次插值样条,再从y方向构造二元有理插值样条曲面;另一方面,按相反次序构造另一个二元有理插值样条曲面;最后将两种插值曲面加权得到一类新的有理插值样条曲面。结果 讨论插值曲面的性质,包括基函数、边界性质、积分加权系数的性质以及误差估计。通过选择合适的参数和加权系数,在不改变插值数据的前提下实现对插值区域内的局部约束控制。结论 实验结果表明,新的加权有理插值样条曲面具有良好的约束控制性质。     

融合残差上采样结构的P2Net无监督单目深度估计

单目深度估计是计算机视觉领域中的一个基本问题, 面片匹配与平面正则化网络(P²Net)是现阶段最先进的无监督单目深度估计方法之一. 由于P²Net中深度预测网络所采用的上采样方法为计算过程较为简单的最近邻插值算法, 使得预测深度图的生成质量较差. 因此, 本文基于多种上采样算法构建出残差上采样结构来替换原网络中的上采样层, 以获取更多特征信息, 提高物体结构的完整性. 在NYU-Depth V2数据集上的实验结果表明, 基于反卷积算法、双线性插值算法和像素重组算法的改进P²Net网络相较原网络在均方根误差RMSE指标上分别降低了2.25%、2.73%和3.05%. 本文的残差上采样结构提高了预测深度图的生成质量, 降低了预测误差.     

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。     

带形状控制的自由曲线曲面参数样条

目的 样条曲线曲面的构造是工程制图中的一个重要部分。针对双曲抛物面上参数样条曲线的构造,在已有的研究基础上提出了一种样条方法使曲线曲面可以任意地逼近一个多边形或者一个网格。方法 在标准四面体内构造一个双曲抛物面,在该曲面上以基函数参数化的方法定义一种带形状参数的参数样条曲线曲面,样条基函数通过将双曲抛物面的有理参数化进行限定,生成单参数有理样条基函数。详细研究了样条的保形性及其端点性质。结果 样条曲线具有一个可变的形状控制因子,可以对曲线进行调整,能以任意精度逼近这个控制四边形或网格。对空间节点列,利用该样条可以生成G²-连续空间曲线,同样对于空间网格可以构造G²-连续的拟合曲面,它所对应的基函数可以是有理形式。结论 实验结果表明,本文在笔者已有的研究基础上提出的参数样条曲线可以通过重心坐标系变换适应为任意的四边形,除了空间四面体内的样条曲线,四面体退化成四边形同样可实现。     

快速收敛的四边形网格三分细分模式

提出了四边形网格的三分细分模式.对于正则和非正则四边形网格,分别采用不同的细分模板获得新的细分顶点.从双三次B样条中推导出正则四边形网格的三分细分模板,极限曲面C²连续;对细分矩阵进行傅里叶变换,推导出非正则四边形网格的三分细分模板,极限曲面C¹连续.提出的三分细分模式可以解决任意拓扑四边形网格的曲面细分问题.与其他细分模式相比,具有收敛速度快、适用范围广等优点.最后给出了四边形网格细分的实例.     

多层次轮廓约束的图像放大算法

为解决图像放大过程中有效地保证边缘锐化的图像插值难题,提出多层次轮廓约束的图像放大算法.首先利用检测算子对图像进行预处理,将图像分为边缘区域、平坦区域;其次,针对图像边缘区域进行自适应梯度扩散获取适当的边缘轮廓层作为图像放大约束;最后对轮廓层直接进行曲线插值重采样,不额外增加边缘层数,以保证放大后的图像在视觉上的边缘清晰.对于非轮廓层的平坦区域,构建双三次Coons插值曲面并进行重采样,保持了平坦区域的平滑性.测试图像为自然图像和医学图像,自然图像的来源是set5和set14测试集,实验对比方法主要从客观效果、视觉效果、时间复杂度3个方面进行比较.实验结果表明,采用该算法得到的放大图像不仅可以保持轮廓清晰,且PSNR及SSIM指标超过了大多数经典的插值算法以及目前流行的基于机器学习的算法.     

一种线性与非线性相结合的图像缩小方法

提出了一种线性和非线性相结合的图像缩小方法.首先由近邻取样方法和邻域平均相结合的方法产生一个中间图像1,利用二元Newton-Thiele型向量连分式建立有理插值曲面;然后对插值曲面进行重新采样产生一个中间图像2;最后对这两幅中间图像进行加权求和,得到最终的图像.实验表明,该方法比常规方法产生的缩小图像效果好.     

Thiele Thiele型二元向量有理插值实现彩色图像的缩放*

提出了一种基于ThieleThiele型有理向量插值的彩色图像插值方法。将数字图像的每一个像素点看成是一个平面域的关于RGB三原色的一个向量,在矩形网格上利用Samelson逆与倒差商技巧,根据图像的像素特征构造ThieleThiele型二元向量连分式有理插值函数,然后对插值曲面进行采样以实现缩放。采用该算法可以得到更加清晰的放大图像。实验结果表明,该方法是一种有效的图像缩放方法。     

Analysis and avoidance of singularities for local G 1 surface interpolation of Bézier curve network with 4-valent nodes

We propose a local method of constructing piecewise G ¹ Bézier patches to span a Bézier curve network with odd- and 4-valent node points. We analyze all possible singular cases of the G ¹ condition that is to be met by the curve network interpolation and propose a new G ¹ continuity condition using linear and quartic scalar weight functions. Using this condition, a curve network can be interpolated without modification at 4-valent nodes with two collinear tangent vectors, even in the presence of singularities. We demonstrate our approach by generating G ¹ surfaces over the curve network which includes singularities at its node vertices and edges.     

三角Bézier曲面的几何约束形变

利用三角Bézier曲面的矩阵表达形式,把几何约束下的形状调整算法从曲线和张量积曲面推广到三角Bézier曲面,使得三角Bézier曲面在形变后既能保持外形大致不变,又能满足一系列事先指定的几何约束(点约束和法向约束).利用Lagange乘子法,几何约束形变的条件极值问题被转化为线性方程组的求解问题,以便于快速计算.特别地,三角Bézier曲面在形变前后还可以满足边界曲线在角点处保持(Ca,Cb,Cc)连续.数值实例表明,该算法简单有效,便于CAD(计算机辅助设计)系统进行交互.     

A Multi‐sided Bézier Patch with a Simple Control Structure

A new n‐sided surface scheme is presented, that generalizes tensor product Bézier patches. Boundaries and corresponding cross‐derivatives are specified as conventional Bézier surfaces of arbitrary degrees. The surface is defined over a convex polygonal domain; local coordinates are computed from generalized barycentric coordinates; control points are multiplied by weighted, biparametric Bernstein functions. A method for interpolating a middle point is also presented. This Generalized Bézier (GB) patch is based on a new displacement scheme that builds up multi‐sided patches as a combination of a base patch, n displacement patches and an interior patch; this is considered to be an alternative to the Boolean sum concept. The input ribbons may have different degrees, but the final patch representation has a uniform degree. Interior control points—other than those specified by the user—are placed automatically by a special degree elevation algorithm. GB patches connect to adjacent Bézier surfaces with G¹continuity. The control structure is simple and intuitive; the number of control points is proportional to those of quadrilateral control grids. The scheme is introduced through simple examples; suggestions for future work are also discussed.     

超分辨率技术的实现——一种改善的小波插值方法

由于成像系统内部和外部的因素 ,使得获取的图象产生退化 .为提高图象的质量 ,发展了一种基于小波插值方法的超分辨率技术 .在简要分析若干典型插值算法的基础上 ,考虑到 Bézier曲面插值方法的精确性和可快速实现性 ,提出了基于 Bézier曲面插值的小波变换方法 .该方法不仅克服了传统插值方法使图象细节退化的缺点 ,同时由于 Bézier曲面插值的快速实现使算法的复杂性得以有效的控制 .实验结果表明了采用 Bézier曲面插值的小波方法对于改善图象分辨率的有效性和可行性 .     

Shape aware normal interpolation for curved surface shading from polyhedral approximation

Independent interpolation of local surface patches and local normal patches is an efficient way for fast rendering of smooth curved surfaces from rough polyhedral meshes. However, the independently interpolating normals may deviate greatly from the analytical normals of local interpolating surfaces, and the normal deviation may cause severe rendering defects when the surface is shaded using the interpolating normals. In this paper we propose two novel normal interpolation schemes along with interpolation of cubic Bézier triangles for rendering curved surfaces from rough triangular meshes. Firstly, the interpolating normal is computed by a Gregory normal patch to each Bézier triangle by a new definition of quadratic normal functions along cubic space curves. Secondly, the interpolating normal is obtained by blending side-vertex normal functions along side-vertex parametric curves of the interpolating Bézier surface. The normal patches by these two methods can not only interpolate given normals at vertices or boundaries of a triangle but also match the shape of the local interpolating surface very well. As a result, more realistic shading results are obtained by either of the two new normal interpolation schemes than by the traditional quadratic normal interpolation method for rendering rough triangular meshes.     

混合有理Bézier曲面及其三维重建

文章将Bernstein基函数与有理Bernstein基函数相结合,构造了一类新型有理曲面-混合有理Bézier曲面;给出了该类曲面的生成方法并讨论了曲面的性质。另一方面,在一种基于Newton-Thiele型非线性方法的插值曲面的三维重建理论基础上,讨论了由离散点集重建混合有理Bézier曲面的问题,为图形图象处理等研究领域提供了新的算法理论。     

Generalized rational Bézier curves for the rigid body motion design

In this paper, we present a new method for the smooth interpolation of the orientations of a rigid body motion. The method is based on the geometrical Hermite interpolation in a hypersphere. However, the non-Euclidean structure of a sphere brings a great challenge to the interpolation problem. For this consideration and the requirements for practical application, we construct the spherical analogue of classical rational Bézier curves, called generalized rational Bézier curves. The new spherical curves are obtained using the generalized rational de Casteljau algorithm, which is a generalization of the classical rational de Casteljau algorithm to a hypersphere. Then, \(G^2\) Hermite interpolation problem in hypersphere is solved analytically using the generalized rational Bézier curve of degree 5. The new method offers residual free parameters including shape parameters and weights, which guarantee the existence of the interpolant to arbitrary motion data and offer great flexibility for the shape design of the motion. Numerical examples show that our method is far better behaved according to the energy functional which is regarded as a measure of the motion shape.     

Preserving computational topology by subdivision of quadratic and cubic Bézier curves

Non-self-intersection is both a topological and a geometric property. It is known that non-self-intersecting regular Bézier curves have non-self-intersecting control polygons, after sufficiently many uniform subdivisions. Here a sufficient condition is given within ℝ³ for a non-self-intersecting, regular C ² cubic Bézier curve to be ambient isotopic to its control polygon formed after sufficiently many subdivisions. The benefit of using the control polygon as an approximant for scientific visualization is presented in this paper.     

Modeling on triangulations with geodesic curves

This paper discusses the problem of modeling on triangulated surfaces with geodesic curves. In the first part of the paper we define a new class of curves, called geodesic Bézier curves, that are suitable for modeling on manifold triangulations. As a natural generalization of Bézier curves, the new curves are as smooth as possible. In the second part we discuss the construction of C ⁰ and C ¹ piecewise Bézier splines. We also describe how to perform editing operations, such as trimming, using these curves. Special care is taken to achieve interactive rates for modeling tasks. The third part is devoted to the definition and study of convex sets on triangulated surfaces. We derive the convex hull property of geodesic Bézier curves.

Flexible G1 interpolation of quad meshes

Transforming an arbitrary mesh into a smooth G¹ surface has been the subject of intensive research works. To get a visual pleasing shape without any imperfection even in the presence of extraordinary mesh vertices is still a challenging problem in particular when interpolation of the mesh vertices is required. We present a new local method, which produces visually smooth shapes while solving the interpolation problem. It consists of combining low degree biquartic Bézier patches with minimum number of pieces per mesh face, assembled together with G¹-continuity. All surface control points are given explicitly. The construction is local and free of zero-twists. We further show that within this economical class of surfaces it is however possible to derive a sufficient number of meaningful degrees of freedom so that standard optimization techniques result in high quality surfaces.