High accuracy geometric Hermite interpolation

We describe a parametric cubic spline interpolation scheme for planar curves which is based on an idea of Sabin for the construction of C¹ bicubic parametric spline surfaces. The method is a natural generalization of [standard] Hermite interpolation. In addition to position and tangent, the curvature is prescribed at each knot. This ensures that the resulting interpolating piecewise cubic curve is twice continuously differentiable with respect to arclength and can be constructed locally. Moreover, under appropriate assumptions, the interpolant preserves convexity and is 6-th order accurate.     

用细分螺线插值容许G2Hermite 数据

为使几何细分方法生成的平面螺线段插值平面容许G2Hermite 数据,基于 平面双圆弧插值理论提出了该方法首末端点处新的细分规则。理论分析表明,修改后的细分 方法所得极限曲线是曲率单调、不变号的螺线段,且插值首末端点处的点、切向、曲率。数 值算例表明,修改后的细分方法收敛速度较快,极限曲线具有较好的形状。     

三次Hermite插值曲线的能量优化

在给定插值点的位置矢量及切矢量的情况下,通过在两相邻节点引入两个新的节点,提出了一类保持[C1]连续的三次Hermite插值曲线的构造方法,分别通过基于曲率、挠率的能量函数对其进行优化,给出了能量最小化的参数取值公式。讨论了参数对曲线形状的影响,实例表明了方法的有效性。     

Hermite interpolation with simple planar PH curves by speed reparametrization

We introduce a new method of solving C¹ Hermite interpolation problems, which makes it possible to use a wider range of PH curves with potentially better shapes. By characterizing PH curves by roots of their hodographs in the complex representation, we introduce PH curves of type K(t−c)²ⁿ⁺¹+d. Next, we introduce a speed reparametrization. Finally, we show that, for C¹ Hermite data, we can use PH curves of type K(t−c)²ⁿ⁺¹+d or strongly regular PH quintics satisfying the G¹ reduction of C¹ data, and use these curves to solve the original C¹ Hermite interpolation problem.     

On Hermite interpolation with B-splines

Polar forms are used to find the B-spline control points for Hermite interpolation.     

Hermite interpolation with a pair of spirals

The Hermite interpolation problem in the plane considered here is to join two points and to match given unit tangent vectors and signed curvatures at the two points with various G² curves consisting of a pair of spirals. The rotation of the tangent vector of the interpolating curve from one point to the other is restricted to being less than π. The necessary and sufficient conditions for the existence of each of the various curves are given.     

基于Hermite插值的SVM研究

在传统SVM的分类求解算法中,由于严格凸的无约束最优化问题中单变量函数x+是不可微的,不能使用通常的最优化的算法进行求解。三次Hermite插值多项式光滑的支持向量机模型采用的是一种多项式光滑技术,用三次Hermite插值多项式代替单变量函数x+,将原来不可微的模型变为可微的模型,并且给出了三次Hermite插值多项式光滑化单变量函数x+的推导过程。使用UCI机器学习数据集中的数据,通过实验验证了该模型的有效性。     

Geometric Hermite interpolation with circular precision

We present several Hermite-type interpolation methods for rational cubics. In case the input data come from a circular arc, the rational cubic will reproduce it.     

Existence and computation of spherical rational quartic curves for Hermite interpolation

n degrees of freedom for any given Hermite data on S ⁿ , n≥2. A method is presented for generating all spherical rational quartic curves on S ⁿ interpolating given Hermite data.     

A note on integer subdivision of NURBS

In a recent article [1], Anantakrishnan and Piegl suggest an approximate method for integer subdivision of rational or nonrational splines. Herein is an analysis of the errors inherent in the approximation.     

基于Hermite插值的网格拼接和融合

网格模型的拼接和融合是3维形状编辑和造型中的一个重要方面。基于Hermite插值技术,提出一种适用于具有一般边界点空间分布的三角网格模型之间无缝光滑拼接和融合方法。首先查找网格模型待拼接区域的边缘点集,并利用二次B样条曲线插值边缘点集分别得到边缘曲线;然后对边缘曲线进行Hermite插值得到拼接区域连续曲面;最后对拼接曲面分别进行三角网格化和Laplacian光顺平滑处理以实现网格模型的光滑拼接和融合。由于利用B样条曲线插值待拼接模型边界,本文方法适用于具有各种不同边界情形的网格模型拼接和融合,它不仅仅可以处理平面边界曲线情形也可以处理空间边界曲线情形。结合Hermite曲面插值拼接过渡区域,使得产生的拼接网格能光滑地衔接待拼接模型。实验结果表明,本文方法能够有效地实现三角网格模型的光滑拼接、模型修复和模型融合。     

Curvature of singular Bézier curves and surfaces

This paper presents a general approach for finding the limit curvature at a singular endpoint of a rational Bézier curve and the singular corner of a rational Bézier surface patch. Conditions for finite Gaussian and mean limit curvatures are expressed in terms of the rank of a matrix.     

Construction of Bézier surface patches with Bézier curves as geodesic boundaries

Given four polynomial or rational Bézier curves defining a curvilinear rectangle, we consider the problem of constructing polynomial or rational tensor-product Bézier patches bounded by these curves, such that they are geodesics of the constructed surface. The existence conditions and interpolation scheme, developed in a general context in earlier studies, are adapted herein to ensure that the geodesic-bounded surface patches are compatible with the usual polynomial/rational representation schemes of CAD systems. Precise conditions for four Bézier curves to constitute geodesic boundaries of a polynomial or rational surface patch are identified, and an interpolation scheme for the construction of such surfaces is presented when these conditions are satisfied. The method is illustrated with several computed examples.     

基于Hermite插值的网格拼接和融合-CIDE2013

网格模型的拼接和融合是3维形状编辑和造型中的一个重要方面。基于Hermite插值技术,提出了一种适用于具有一般边界点空间分布的三角网格模型之间无缝光滑拼接和融合方法。首先查找网格模型待拼接区域的边缘点集,并利用二次B样条曲线插值边缘点集分别得到边缘曲线;然后对边缘曲线进行Hermite插值得到拼接区域连续曲面;最后对拼接曲面分别进行三角网格化和Laplacian光顺平滑处理以实现网格模型的光滑拼接和融合。由于利用B样条曲线插值待拼接模型边界,本文方法适用于具有各种不同边界情形的网格模型拼接和融合,它不仅仅可以处理平面边界曲线情形也可以处理空间边界曲线情形。结合Hermite曲面插值拼接过渡区域,使得产生的拼接网格能光滑地衔接待拼接模型。实验结果表明,本文方法能够有效地实现三角网格模型的光滑拼接、模型修复和模型融合。     

Boundary evaluation for interval Bézier curve

The objective of this paper is to provide an efficient and reliable algorithm for representing and evaluating the boundary of the interval Bézier curve in 2- and 3-D. The boundary of the planar Bézier curve is represented by a sequence of Bézier curve segments with same degree and line segments in the order they are encountered when marching counter-clockwise along its boundary. The boundary can also be represented as a single B-spline curve having the same degree with the interval Bézier curve. The boundary of the 3-D interval Bézier curve is made up of trimmed Bézier surface patches and rectangular patches. Some examples illustrate our algorithms.     

基于Hermite插值的视频图像分辨率匹配电路设计

目前常用的图像分辨率匹配算法,存在着缩放效果模糊或硬件消耗过大等问题。综合缩放效果和电路消耗考虑,提出了一种基于三次Hermite插值算法的图像分辨率匹配电路。以基于数字差分分析算法的模块控制插值源像素的选取,以Hermite算法为内核完成插值系数的计算,相对双三次插值算法,保证了较小的硬件消耗。通过均方差、峰值信噪比、灰度梯度模值和傅里叶频谱对Hermite插值算法缩放效果与其余算法缩放效果进行定量分析,表明三次Hermite插值算法比双线性插值算法具有更好的缩放效果。同时通过电路综合结果,表明该电路相对基于双三次插值的电路,硬件消耗更小。     

Non-self-overlapping Hermite interpolation mapping: a practical solution for structured quadrilateral meshing

This paper addresses the problem of constructing a structured quadrilateral grid inside a given four-sided 2D region by a particular boundary-conforming mapping scheme—Hermite Interpolation Mapping (HIM). When the four given boundary curves are concave and convoluted, all boundary-conform mapping methods suffer from potential self-overlapping problem. Under HIM, the geometry of the grid depends on both the four boundary curves and the tangent vector functions associated with the curves. While the four boundary curves are fixed, the tangent functions in HIM can be varied to suit the need of controlling the characteristics of the mesh inside the given region to prevent self-overlapping. Besides tangent functions, the four twist vectors at the corners of the region can also be adjusted to influence the distribution of the inner grid elements. In our approach, a constrained functional optimization scheme is adopted to adjust the tangent functions and the twist vectors, adaptive to the geometry of the boundary curves, so that the resulting HIM will be free of self-overlapping. The optimization is carried out on the shape control energy that measures the overall mesh quality of the underlying HIM while the self-overlapping is strongly prevented in the form of constraints to the optimization. Experimental results show the promise of the proposed method as a practical and effective solution for structured grid generation.