LSM: A layer subdivision method for deformable object matching

A deformation technique is a method to deform any part of, or an entire object, into a desired shape. Existing deformation methods take a lot of computational cost to represent smoothness correctly due to the constraints caused by differential coefficients of high degree. Thus, it is very difficult to find a general solution. In this paper we propose a LSM (layered subdivision method) that integrates a controlling mechanism, surface deformation, and mesh refinement processing 3D modeling and free-form deformable object matching. The proposed method is considerably more efficient and robust when compared to the existing method of free-form surface, because of the computation of the reference points of deformation edge using geometry of free-form surface. This approach can be applied to automatic inspection of NURBS models and object recognition.     

基于控制顶点扰动的平面Offset曲线的NURBS逼近

平面曲线的ｏｆｆｓｅｔ曲线具有丰富的几何结构,它在曲面造型、ＮＣ加工等领域具有广泛应用,但除直线、圆弧或速端曲线等少数几种曲线外,有理多项式参数曲线的ｏｆｆｓｅｔ曲线不能保证仍是有理多项式曲线形式。因此,实际应用中常用逼近方法表示ｏｆｆｓｅｔ曲造型系统中数据结构和几何算法的统一表示。作者针对平面ＮＵＲＢＳ曲线的特点,提出两种逼近表示方法,一种是基于曲线分割的控制顶点动法,另一种是整体控制顶点偏移法     

A Geometric Representation Scheme Suitable for Shape Optimization

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二次NURBS 曲线的退化曲线

NURBS 曲线是几何造型中广泛使用的曲线拟合工具。当某一权因子趋向于无穷 时,NURBS 曲线趋于相应的控制顶点,当所有权因子趋向于无穷时,其极限曲线的几何性质 目前还没有结论。利用NURBS 曲线的节点插入算法,将NURBS 曲线转化为分段有理Bézier 曲线,结合有理Bézier 曲线的退化理论,得到当所有权因子趋向于无穷时其退化曲线的几何 结构。     

平面参数曲线的自由变形

提出一种全新的参数曲线自由变形方法:基于平面矢量叠加原理构造出特殊的伸缩矢量函数去作用待变形曲线,从而使曲线发生形变,通过改变曲线上型值点的位置和个数以及每个型值点处的权因子可达到预期的变形效果.实验表明,该方法数学背景简单、易于控制,重复使用可获得丰富的变形效果,适用于几何造型和计算机动画等领域.     

Multi-degree reduction of NURBS curves based on their explicit matrix representation and polynomial approximation theory

NURBS curve is one of the most commonly used tools in CAD systems and geometric modeling for its various specialties, which means that its shape is locally adjustable as well as its continuity order, and it can represent a conic curve precisely. But how to do degree reduction of NURBS curves in a fast and efficient way still remains a puzzling problem. By applying the theory of the best uniform approximation of Chebyshev polynomials and the explicit matrix representation of NURBS curves, this paper gives the necessary and sufficient condition for degree reducible NURBS curves in an explicit form. And a new way of doing degree reduction of NURBS curves is also presented, including the multi-degree reduction of a NURBS curve on each knot span and the multi-degree reduction of a whole NURBS curve. This method is easy to carry out, and only involves simple calculations. It provides a new way of doing degree reduction of NURBS curves, which can be widely used in computer graphics and industrial design.     

Optimized GPU evaluation of arbitrary degree NURBS curves and surfaces

This paper presents a new unified and optimized method for evaluating and displaying trimmed NURBS surfaces using the Graphics Processing Unit (GPU). Trimmed NURBS surfaces, the de facto standard in commercial mechanical CAD modeling packages, are currently being tessellated into triangles before being sent to the graphics card for display since there is no native hardware support for NURBS. Other GPU-based NURBS evaluation and display methods either approximated the NURBS patches with lower degree patches or relied on specific hard-coded programs for evaluating NURBS surfaces of different degrees. Our method uses a unified GPU fragment program to evaluate the surface point coordinates of any arbitrary degree NURBS patch directly, from the control points and knot vectors stored as textures in graphics memory. This evaluated surface is trimmed during display using a dynamically generated trim-texture calculated via alpha blending. The display also incorporates dynamic Level of Detail (LOD) for real-time interaction at different resolutions of the NURBS surfaces. Different data representations and access patterns are compared for efficiency and the optimized evaluation method is chosen. Our GPU evaluation and rendering speeds are more than 40 times faster than evaluation using the CPU.     

三维试衣系统中的H-Anim格式人体模型构造

介绍了一种容易实现的、实用性强的人体模型构造方法，可以广泛应用于服装的三维试衣系统或其他CAD领域。根据用户输入的参数或者交互式的操作，对一个标准的VRML H-Anim 1．1格式人体模型先进行骨架变形，再使用水平和垂直两条NURBS曲线控制身体各部分的变形，从而达到用户需要的效果。     

BBN曲面的形状分析与控制

研究了调节凸Bézier曲面、B-样条曲面及NURBS曲面（BBN曲面）一个控制点以后,曲面形状变化的规律.通过将BBN曲面分解成一张凸曲面与具有特殊形状曲面的叠加,建立了曲面变形前后一些几何量与变形位移量之间的数量关系,得到了凸BBN曲面失去凸性的充分条件和判据.相应的结果可应用于调节与控制BBN曲面形状的算法设计.     

Local T-spline surface skinning

Skinning or lofting remains a challenging problem in computer graphics and free-form surface design. Although it was addressed by many researchers, no sufficiently general solution has been proposed yet. In the interpolating approach, the incompatibility of the input NURBS curves are solved by knot insertion. This process leads to an explosion in the number of control points defining the skinned surface. Other methods avoid this problem by generating skinned surfaces that approximate rather than interpolate the input curves. In this paper, we provide a solution to this problem using T-splines. Compared with existing approaches, a T-spline skinned surface interpolates a set of incompatible curves with a control mesh of fewer vertices. Typically, the linear system involved could be solved globally. However, our approach provides a local solution for each skinned curve. As such, local modification could be used to meet additional constraints such as given normal and/or predefined curvature across the skinned curves.     

NURBS曲线拟合的最小二乘渐进迭代逼近优化算法

为了使NURBS曲线更精确地拟合散乱数据点,提出了一种基于最小二乘渐进迭代逼近(least square progressive and iterative approximation,LSPIA)的NURBS曲线拟合优化算法.首先,确定一条初始NURBS曲线,利用LSPIA算法优化控制顶点;然后,分别优化数据点参数,拟合曲线的节点和权因子,每优化好一个变量,重新优化控制顶点;最后,经多次优化迭代得到高精度的NURBS拟合曲线.在优化每类变量时,为了避免被其他变量影响,保持其他变量不变.基于LSPIA的NURBS曲线拟合优化算法充分利用了LSPIA算法的优点,在迭代过程中,可以重复使用前一迭代步骤得到的控制顶点等数据,从而节省了运算时间.算法实例表明,该算法能获得一定保形效果.     

Interactive Rendering of Deforming NURBS Surfaces

Non-uniform rational B-splines (NURBS) has been widely accepted as a standard tool for geometry representation and design. Its rich geometric properties allow it to represent both analytic shapes and free-form curves and surfaces precisely. Moreover, a set of tools is available for shape modification or more implicitly, object deformation. Existing NURBS rendering methods include de Boor algorithm, Oslo algorithm, Shantz’s adaptive forward differencing algorithm and Silbermann’s high speed implementation of NURBS. However, these methods consider only speeding up the rendering process of individual frames. Recently, Kumar et al. proposed an incremental method for rendering NURBS surfaces, but it is still limited to static surfaces. In real-time applications such as virtual reality, interactive display is needed. If a virtual environment contains a lot of deforming objects, these methods cannot provide a good solution. In this paper, we propose an efficient method for interactive rendering of deformable objects by maintaining a polygon model of each deforming NURBS surface and adaptively refining the resolution of the polygon model. We also look at how this method may be applied to multi-resolution modelling.     

参数曲线曲面自由变形的多项式因子方法

为得到理想的造型效果,提出一种空间参数曲线曲面自由变形的方法.首先引入基于多项式的伸缩因子,并构造了空间变形矩阵;然后将变形矩阵或伸缩因子作用于待变形曲线曲面方程,从而得到形变效果.实验结果表明,该方法计算简单、易于控制,可得到较好的变形效果.     

渐进迭代逼近方法在曲线变形上的应用

目的 随着几何造型、计算机动画等领域的快速发展,曲线的自由变形技术在近年来受到了广泛的关注。为了获得更多有趣、逼真的变形效果,提出基于渐进迭代逼近与主顶点方法的曲线局部变形算法。方法 给定数据点集,首先采用渐进迭代逼近方法或是基于最小二乘的渐进迭代逼近方法产生待变形曲线;其次对待变形区域使用延拓准则,基于主顶点方法与待变形曲线的形状信息选取控制顶点进行调整;最后对调整后的控制顶点运用局部渐进迭代逼近方法生成逼近曲线,得到期望的变形效果。结果 此变形操作借助于局部渐进迭代逼近方法,具有较好的灵活性。通过茶壶、面部轮廓、手等数值实例,表明了该方法可以得到良好的变形效果。进一步地,借助于叠加变形还可以得到整体的、周期的、伸缩的等各类更加丰富的变形效果。结论 本文研究渐进迭代逼近在曲线变形上的应用,将主顶点方法引入曲线的变形之中,把两者相结合提出了基于渐进迭代逼近与主顶点方法的曲线局部变形算法。该算法不仅具备渐进迭代逼近方法的收敛稳定性,且借助于主顶点方法,可以得到较好的变形效果。该方法适用于曲线的局部变形,丰富了曲线的变形效果。     

Free-Form Deformation with Rational DMS-Spline Volumes

In this paper, we propose a novel free-form deformation （FFD） technique, RDMS-FFD （Rational DMS-FFD）, based on rational DMS-spline volumes. RDMS-FFD inherits some good properties of rational DMS-spline volumes and combines more deformation techniques than previous FFD methods in a consistent framework, such as local deformation, control lattice of arbitrary topology, smooth deformation, multiresolution deformation and direct manipulation of deformation. We first introduce the rational DMS-spline volume by directly generalizing the previous results related to DMS-splines. How to generate a tetrahedral domain that approximates the shape of the object to be deformed is also introduced in this paper. Unlike the traditional FFD techniques, we manipulate the vertices of the tetrahedral domain to achieve deformation results. Our system demonstrates that RDMS-FFD is powerful and intuitive in geometric modeling.     

Neural Network Methods for NURBS Curve and Surface Interpolation

New algorithms based on artificial neural network models are presented for cubic NURBS curve and surface interpolation.When all th knot spans are identical,the NURBS curve interpolation procedure degenerates into that of uniform rational B-spline curves.If all the weights of data points are identical,then the NURBS curve interpolation procedure degenerates into the integral B-spline curve interpolation.     

Modeling 3D Curves of Minimal Energy

Modeling a curve through minimizing its energy yields an overall smooth curve. A common way to model shape features is to perform the minimization subject to a number of interpolation constraints. This way of modeling is attractive because the designer is not bothered with the precise representation of the curve (e.g. control points). However, local shape specification by means of interpolation constraints is very limited. On the other hand, local deformation by repositioning control points is powerful but very laborious, and destroys the minimal energy property. In this paper, deform operators are introduced for 3D curve modeling that have built-in energy terms that have an intuitive effect. These operators allow local shape modification and do justice to the energy minimization way of modeling.     

一种基于快速拟合的NURBS曲面实体建模方法

为了解决实体对象建模时出现的失真性和控制点膨胀的问胚,提出了一种基于NURBS建模的快速拟合算法,算法结合所设计的应用型曲面实体数据结构,针对扫掠实体、直纹实体、蒙皮实体和旋转实体,分别采用整合法、双基面互导求均法、平均法和旋转因子法来计算形成实体所需的最少控制点和W方向节点矢量,将控制点的计算从三维抽象到二维.实验证明算法不仅有效解决了失真性和控制点膨胀的问题,而且还具有通过调节幂次来实现用户的不同需要和运算速度快等优点.     

Full analytical sensitivities in NURBS based isogeometric shape optimization

Non-uniform rational B-spline (NURBS) has been widely used as an effective shape parameterization technique for structural optimization due to its compact and powerful shape representation capability and its popularity among CAD systems. The advent of NURBS based isogeometric analysis has made it even more advantageous to use NURBS in shape optimization since it can potentially avoid the inaccuracy and labor-tediousness in geometric model conversion from the design model to the analysis model.Although both positions and weights of NURBS control points affect the shape, until very recently, usually only control point positions are used as design variables in shape optimization, thus restricting the design space and limiting the shape representation flexibility.This paper presents an approach for analytically computing the full sensitivities of both the positions and weights of NURBS control points in structural shape optimization. Such analytical formulation allows accurate calculation of sensitivity and has been successfully used in gradient-based shape optimization.The analytical sensitivity for both positions and weights of NURBS control points is especially beneficial for recovering optimal shapes that are conical e.g. ellipses and circles in 2D, cylinders, ellipsoids and spheres in 3D that are otherwise not possible without the weights as design variables.