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.     

Geometric algorithm for point projection and inversion onto Bézier surfaces

This paper presents an accurate and efficient method for the computation of both point projection and inversion onto Bézier surfaces. First, these two problems are formulated in terms of solution of a polynomial equation with u and v variables expressed in the Bernstein basis. Then, based on subdivision of the Bézier surface and the recursive quadtree decomposition, a novel solution method is proposed. The computation of point projection is shown to be equivalent to the geometrically intuitive intersection of asurface with the u-ν plane. Finally, by comparing the distances between the test point and the candidate points, the closest point is found. Examples illustrate the feasibility of this method.     

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 .     

A systematic framework for on‐line calibration of a head‐mounted projection display for augmented‐reality systems

Abstract— Augmented reality (AR) is a technology in which computer‐generated virtual images are dynamically superimposed upon a real‐world scene to enhance a user's perceptions of the physical environment. A successful AR system requires that the overlaid digital information be aligned with the user's real‐world senses — a process known as registration. An accurate registration process requires the knowledge of both the intrinsic and extrinsic parameters of the viewing device and these parameters form the viewing and projection transformations for creating the simulations of virtual images. In our previous work, an easy off‐line calibration method in which an image‐based automatic matching method was used to establish the world‐to‐image correspondences was presented, and it is able to achieve subpixel accuracy. However, this off‐line method yields accurate registration only when a user's eye placements relative to the display device coincides with locations established during the offline calibration process. A likely deviation of eye placements, for instance, due to helmet slippage or user‐dependent factors such as interpupillary distance, will lead to misregistration. In this paper, a systematic on‐line calibration framework to refine the off‐line calibration results and to account for user‐dependent factors is presented. Specifically, based on an equivalent viewing projection model, a six‐parameter on‐line calibration method to refine the user‐dependent parameters in the viewing transformations is presented. Calibration procedures and results as well as evaluation experiments are described in detail. The evaluation experiments demonstrate the improvement of the registration accuracy.     

基于扩展Bézier曲线拼接的曲线造型新方法

提出了一种基于扩展Bézier曲线拼接的曲线造型新方法。该方法首先构造了一种具有优良形状可调性和更好逼近性的带3个形状参数α, β, γ的三次扩展Bézier曲线(CE-Bézier曲线);并针对CE-Bézier曲线无法精确表示圆弧和椭圆弧等二次曲线的缺点,利用CE-Bézier曲线与C-Bézier曲线间的拼接技术,解决了CE-Bézier曲线造型中圆弧和椭圆弧的表示问题。最后讨论了该方法在曲线曲面设计中的应用。造型实例表明,该方法在计算机辅助几何设计中具有一定的应用价值。     

一种基于Bézier插值曲面的图像放大方法

文章提出了一种利用Bézier插值曲面进行图像放大的方法,该方法是为数字图像的每一个色彩分量构造一个分块双三次Bzier插值C¹曲面,图像放大等价于以不同的采样速率对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.     

三角域上拟二次Bézier曲面片的构造及其应用

针对计算机辅助几何设计中三角曲面片造型方法进行了研究。在非多项式空间中构造了一组基函数,分析了该基函数的性质;利用七个控制顶点定义了相应的三角曲面片,由于该三角曲面片具有类似于三角域上二次Bézier曲面片的性质,故称其为拟二次Bézier三角曲面片;举例说明了拟二次Bézier三角曲面片不仅边界可以精确表示圆弧和椭圆弧,而且可以通过多引入的一个控制顶点实现在边界保持不变的情况下对曲面形状进行调节,同时,该曲面片可作为过渡曲面在三通管造型接口处实现光滑过渡。总之,拟二次Bézier三角曲面片在曲面造型与曲面设计中有较好的应用,可作为现有造型方法的有效补充。     

Wavelet Rasterization

We present a method for analytically calculating an anti‐aliased rasterization of arbitrary polygons or fonts bounded by Bézier curves in 2D as well as oriented triangle meshes in 3D. Our algorithm rasterizes multiple resolutions simultaneously using a hierarchical wavelet representation and is robust to degenerate inputs. We show that using the simplest wavelet, the Haar basis, is equivalent to performing a box‐filter to the rasterized image. Because we evaluate wavelet coefficients through line integrals in 2D, we are able to derive analytic solutions for polygons that have Bézier curve boundaries of any order, and we provide solutions for quadratic and cubic curves. In 3D, we compute the wavelet coefficients through analytic surface integrals over triangle meshes and show how to do so in a computationally efficient manner.     

基于微粒群算法的有理Bézier曲线降阶

从最优化思想出发，把有理Bézier曲线的降阶问题转化为求解优化问题，并基于微粒群算法，给出有理Bézier曲线降阶的一种新方法。该方法可以实现多次降阶，且降阶后的有理Bézier曲线直接以显式给出。最后结合实例，与使用遗传算法进行有理Bézier曲线降阶的结果进行对比，实验结果表明了微粒群算法的有效性。     

Exaggerating Character Motions Using Sub‐Joint Hierarchy

Motion capture cannot generate cartoon‐style animation directly. We emulate the rubber‐like exaggerations common in traditional character animation as a means of converting motion capture data into cartoon‐like movement. We achieve this using trajectory‐based motion exaggeration while allowing the violation of link‐length constraints. We extend this technique to obtain smooth, rubber‐like motion by dividing the original links into shorter sub‐links and computing the positions of joints using Bézier curve interpolation and a mass‐spring simulation. This method is fast enough to be used in real time.     

Texture Synthesis From Photographs

The goal of texture synthesis is to generate an arbitrarily large high‐quality texture from a small input sample. Generally, it is assumed that the input image is given as a flat, square piece of texture, thus it has to be carefully prepared from a picture taken under ideal conditions. Instead we would like to extract the input texture from any surface from within an arbitrary photograph. This introduces several challenges: Only parts of the photograph are covered with the texture of interest, perspective and scene geometry introduce distortions, and the texture is non‐uniformly sampled during the capture process. This breaks many of the assumptions used for synthesis. In this paper we combine a simple novel user interface with a generic per‐pixel synthesis algorithm to achieve high‐quality synthesis from a photograph. Our interface lets the user locally describe the geometry supporting the textures by combining rational Bézier patches. These are particularly well suited to describe curved surfaces under projection. Further, we extend per‐pixel synthesis to account for arbitrary texture sparsity and distortion, both in the input image and in the synthesis output. Applications range from synthesizing textures directly from photographs to high‐quality texture completion.     

Binary contour coding using Bézier approximation

A method for coding of binary image contour using Bézier approximation is proposed. A set of key pixel (guiding pixels) on the contour is defined which enables the contour to be decomposed into arcs and straight line segments. A set of cleaning operations has been considered as an intermediate step before producing the final output.The quality of fauthful reproduction of the decoded verions has been examined through the objective measures of shape compactness and the percentage error in area. Finally, the bit requirement and ther compression ratios for different input images are compared with the existing ones.     

Optical simulation for cross‐talk evaluation and improvement of autostereoscopic 3‐D displays with a projector array

Abstract— Multi‐view spatial‐multiplexed autostereoscopic 3‐D displays normally use a 2‐D image source and divide the pixels to generate perspective images. Due to the reduction in the resolution of each perspective image for a large view number, a super‐high‐resolution 2‐D image source is required to achieve 3‐D image quality close to the standard of natural vision. This paper proposes an approach by tiling multiple projection images with a low magnification ratio from a microdisplay to resolve the resolution issue. Placing a lenticular array in front of the tiled projection image can lead to an autostereoscopic display. Image distortion and cross‐talk issues resulting from the projection lens and pixel structure of the microdisplay have been addressed with proper selection of the active pixel and adequate pixel grouping and masking. Optical simulation has shown that a 37‐in. 12‐view autostereoscopic display with a full‐HD (1920 × 1080) resolution can be achieved with the proposed 3‐D architecture.     

Bézier样条曲线改进的近似弧长参数化方法

提出了Bézier样条曲线利用分割技术近似弧长参数化的一种方法,并给出了相应的算法。通过求出曲线上所谓的‘最坏点’并在相应点处进行分割,可得到两条Bézier样条曲线。让这两条Bézier样条曲线具有与它们的近似弧长成比例的权,并对所得到的新的Bézier样条曲线进行同样的工作最终可得到一条由多条Bézier样条曲线所构成的新曲线。将这多条Bézier样条曲线合并成为一条Bézier样条曲线并通过节点插入技术将所得Bézier样条曲线转化为B-样条曲线的形式可得到全局参数域,其中各条Bézier曲线在全局参数域中所占子区间的长度与它们的权成比例,这样便得到了一条近似弧长参数化曲线。     

Geometric algorithm for point projection and inversion onto Bézier surfaces

This paper presents an accurate and efficient method for the computation of both point projection and inversion onto Bézier surfaces. First, these two problems are formulated in terms of solution of a polynomial equation with u and v variables expressed in the Bernstein basis. Then, based on subdivision of the Bézier surface and the recursive quadtree decomposition, a novel solution method is proposed. The computation of point projection is shown to be equivalent to the geometrically intuitive intersection of a surface with the u-v plane. Finally, by comparing the distances between the test point and the candidate points, the closest point is found. Examples illustrate the feasibility of this method.     

基于多分辨率模型的三角曲面特征线辨识技术

由于特征线在反求工程 CAD建模中具有非常重要的作用 ,因此利用图形图象处理中的多分辨率模型概念 ,通过研究三角曲面模型的特征线 ,提出了一种三角曲面特征线的计算方法 .将计算得到的初始特征线通过编辑、修改等手段进行处理 ,得到清晰的特征线 ,并将其作为进一步划分重构 B样条曲面边界的依据和参考 ,从而为实现基于三角曲面模型的 B样条曲面重构奠定了基础 .实验结果证明 ,该算法能够在三角曲面上提取出令人满意的特征线 ,并据此重构出拓扑划分合理的 B样条曲面 .     

PDE曲面的Bézier逼近

为了实现PDE(partial differential equation)曲面造型技术与传统CAD(computer aided design)造型系统的数据交换,基于约束优化的思想,给出了PDE曲面的Bézier逼近算法,并利用张量积Bézier曲面的细分性质对该算法进行了优化.所给出的计算实例及误差比较结果说明了该算法的有效性.     

Bézier曲面的函数复合及其应用

目前有两种常用的Bézier曲面片,分别称为三角和四边Bézier曲面片,它们分别用不同的基函数表示.本文通过移位算子和函数复合的方法,得到了两个关于这两种Bézier曲面片的结果.一个是四边Bézier曲面片与一次三角Bézier函数的复合,另一个是三角Bézier曲面片与双线性四边Bézier函数的复合.在每一种情况中,复合所得到的Bézier曲面片的控制顶点是原来Bézier曲面片的控制顶点的线性组合.移位算子的应用使得相应的推导过程变得简洁和直观.这两个结果的应用包括：两种Bézier面片间的转化     

Subdivision algorithms for Bézier triangles

In an earlier paper the author studied subdivision algorithms for Bézier curves. These algorithms were generated by constructing degenerate Bézier triangles and tetrahedra and using their edges to subdivide the original curve. In this paper we take up the study of subdivision algorithms for the triangular Bézier surfaces first introduced by Sabin. It will be shown that such algorithms can be generated by constructing higher dimensional degenerate Bézier simplices and using their triangular subsimplices to subdivide the original triangular surface.     

High-Performance Computation of Bézier Surfaces on Parallel and Heterogeneous Platforms

Bézier surfaces are mathematical tools employed in a wide variety of applications. Some works in the literature propose parallelization strategies to improve performance for the computation of Bézier surfaces. These approaches, however, are mainly focused on graphics applications and often are not directly applicable to other domains. In this work, we propose a new method for the computation of Bézier surfaces, together with approaches to efficiently map the method onto different platforms (CPUs, discrete and integrated GPUs). Additionally, we explore CPU–GPU cooperation mechanisms for computing Bézier surfaces using two integrated heterogeneous systems with different characteristics. An exhaustive performance evaluation—including different data-types, rendering and several hardware platforms—is performed. The results show that our method achieves speedups as high as 3.12x (double-precision) and 2.47x (single-precision) on CPU, and 3.69x (double-precision) and 13.14x (single-precision) on GPU compared to other methods in the literature. In heterogeneous platforms, the CPU–GPU cooperation increases the performance up to 2.09x with respect to the GPU-only version. Our method and the associated parallelization approaches can be easily employed in domains other than computer-graphics (e.g., image registration, bio-mechanical modeling and flow simulation), and extended to other Bézier formulations and Bézier constructions of higher order than surfaces.