Bézier 曲线的多项式重新参数化检测

研究了一种用于精确检测一条Bézier 曲线的次数是否可以通过多项式重新参数化 降低的算法。该算法对任意一条Bézier 曲线,将重新参数化前后的基函数的关系用方程组的形 式表达,但不需要解方程,而是通过系数表示的金字塔算法直接计算,可以精确求出用于重新 参数化的多项式和降低次数后的Bézier 曲线的控制顶点,并且该重新参数化的多项式在相差一 个线性变换的前提下是唯一的。通过实例应用,该算法运算速度较之前的算法快。     

基于四阶贝塞尔曲线的无人车可行轨迹规划

对于实际的无人车系统来说,轨迹规划需要保证其规划出来的轨迹满足运动学约束、 侧滑约束以及执行机构约束.为了生成满足无人车初始状态约束、目标状态约束的局部可行轨迹,本文提出了一种基于四阶贝塞尔曲线的轨迹规划方法.在该方法中, 轨迹规划问题首先被分解为轨形规划及速度规划两个子问题.为了满足运动学约束、 初始状态约束、目标状态约束以及曲率连续约束,本文采用由3个参数确定的四阶贝塞尔曲线来规划轨迹形状.为了保证转向机构可行,本文进一步采用优化方法求解一组最优参数从而规划出曲率变化最小的轨线.对于轨线执行速度规划,为了满足速度连续约束、加速度连续约束、加速度有界约束以及目标状态侧滑约束,本文首先求解了可行的轨迹执行耗时区间,再进一步在该区间中求解能够保证任意轨迹点满足侧滑约束的耗时,最后再由该耗时对任意点速度进行规划.本文结合实际无人车的应用对轨迹搜索空间生成、道路行车模拟以及路径跟踪进行了仿真实验,并基于实际的环境数据进行了轨迹规划实验.     

Real-Time GPU Silhouette Refinement using Adaptively Blended Bézier Patches

We present an algorithm for detecting and extracting the silhouette edges of a triangle mesh in real time using Graphical Processing Units (GPUs). We also propose a tessellation strategy for visualizing the mesh with smooth silhouettes through a continuous blend between Bézier patches with varying level of detail. Furthermore, we show how our techniques can be integrated with displacement and normal mapping. We give details on our GPU implementation and provide a performance analysis with respect to mesh size.     

Energy Duality Methods for Piecewise Bézier Curve Construction

Piecewise Bézier Curves are constructed using a minimization principle. C^k and GC^k continuity is imposed by linear constraints. The corresponding quadratic programming with linear constraints problem is introduced and solved by duality methods. Bordering matrices methods are implemented to deal with local refinement (subdivision). The result is a versatile tool for defining/editing contours made of piecewise Bézier curves.     

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.     

A shape controled fitting method for Bézier curves

The problem of controling a shape when fitting a curve to a set of digitized data points by proceeding to a least squares approximation is considered. A nonlinear method of solving this problem, dedicated to the obtention of planar curves with a smooth and monotonous variation of curvature is introduced. This method uses particular Bézier curves, called typical curves, whose control polygon is partially constrained in order to provide the desired curve shape. The curve fitting principle is based on variations of the tangent direction at the ends of the curve. These variations are controled by the displacement of a given curve point. An automatic procedure using this method to get a curve close to a set of data points has been implemented. An application to car body shape design and a comparison with the least squares approximation method is presented and discussed.     

Progressive iterative approximation for triangular Bézier surfaces

Recently, for the sake of fitting scattered data points, an important method based on the PIA (progressive iterative approximation) property of the univariate NTP (normalized totally positive) bases has been effectively adopted. We extend this property to the bivariate Bernstein basis over a triangle domain for constructing triangular Bézier surfaces, and prove that this good property is satisfied with the triangular Bernstein basis in the case of uniform parameters. Due to the particular advantages of triangular Bézier surfaces or rational triangular Bézier surfaces in CAD (computer aided design), it has wide application prospects in reverse engineering.     

A Bernstein-Bézier Based Approach to Soft Tissue Simulation

This paper discusses a Finite Element approach for volumetric soft tissue modeling in the context of facial surgery simulation. We elaborate on the underlying physics and address some computational aspects of the finite element discretization.
In contrast to existing approaches speed is not our first concern, but we strive for the highest possible accuracy of simulation. We therefore propose an extension of linear elasticity towards incompressibility and nonlinear material behavior, in order to describe the complex properties of human soft tissue more accurately. Furthermore, we incorporate higher order interpolation functions using a Bernstein-Bézier formulation, which has various advantageous properties, such as its integral polynomial form of arbitrary degree, efficient subdivision schemes, and suitability for geometric modeling and rendering. In addition, the use of tetrahedral Finite Elements does not put any restriction on the geometry of the simulated volumes.
Experimental results obtained from a synthetic block of soft tissue and from the Visible Human Data Set illustrate the performance of the envisioned model.     

基于二项式系数设计矩阵的Bézier曲线扩展

根据二项式的展开系数,设计出带形状参数的正系数矩阵,并对Bernstein基函数进 行具有明显几何意义的构造,推导出同阶带参的A-Bernstein基函数,该基函数具有Bernstein基函 数类似的性质。在此基础上推导出对应的A-Bézier曲线,分析了其不但具有Bézier曲线类似的性质, 而且在原始控制点不变的情况下,可以通过修改形状参数来对曲线进行调整。此外,还进一步说 明了可以通过对正系数矩阵的调整,实现对曲线的调整。通过举例,展现出该方法灵活有效。     

Multi-degree reduction of Bézier curves using reparameterization

L₂-norms are often used in the multi-degree reduction problem of Bézier curves or surfaces. Conventional methods on curve cases are to minimize , where and are the given curve and the approximation curve, respectively. A much better solution is to minimize , where is the closest point to point , that produces a similar effect as that of the Hausdorff distance. This paper uses a piecewise linear function L(t) instead of t to approximate the function φ(t) for a constrained multi-degree reduction of Bézier curves. Numerical examples show that this new reparameterization-based method has a much better approximation effect under Hausdorff distance than those of previous methods.     

A comparison of local parametric C Bézier interpolants for triangular meshes

Parametric curved shape surface schemes interpolating vertices and normals of a given triangular mesh with arbitrary topology are widely used in computer graphics for gaming and real-time rendering due to their ability to effectively represent any surface of arbitrary genus. In this context, continuous curved shape surface schemes using only the information related to the triangle corresponding to the patch under construction, emerged as attractive solutions responding to the requirements of resource-limited hardware environments. In this paper we provide a unifying comparison of the local parametric C⁰ curved shape schemes we are aware of, based on a reformulation of their original constructions in terms of polynomial Bézier triangles. With this reformulation we find a geometric interpretation of all the schemes that allows us to analyse their strengths and shortcomings from a geometrical point of view. Further, we compare the four schemes with respect to their computational costs, their reproduction capabilities of analytic surfaces and their response to different surface interrogation methods on arbitrary triangle meshes with a low triangle count that actually occur in their real-world use.     

A Viewpoint-Dependent Approach to Ray Trace Free-Form Surfaces

For convincing realistic scenes objects with free‐form surfaces are essential. Especially for photorealistic rendering pure polygonal models are often not sufficient. We present a new kind of algorithm to render free‐form surfaces in a rendering system based on ray tracing. We describe a triangular patch as usual by its three points and normal vectors, but base the intersection calculation as well on the viewpoint of the camera (or, in general, on the ray itself). Hence, the shape of the object depends to some extent on the sampling rays. However, the resulting differences of, for instance, the shape of the silhouette to the shape of the corresponding shadow is usually not perceived by the observer of the rendered image. Because we perform a direct computation without a tessellation process, the resulting surface, its shadows, and its reflections appear smooth independent of the distance to the camera. Furthermore, the memory consumption depends only linearly on the number of input triangles. Special features like creases, T‐vertices, and darts are also well supported. The computed uv‐coordinates provide a direct means for texture mapping whose visual appearance improves significantly compared to triangle meshes of the same resolution.     

一种分层海洋中求取本征声线的新方法

提出了一种在恒定深度的分层海洋中计算本征声线的方法。该方法在跨度模型的基础上，将本征声线表示为三段声线跨度的组合，建立跨度方程组以确定本征声线。采用计算本征声线的方法，可以计算若干条本征声线的出射角、到达角、到达时间和传播损失。如果给定声源的辐射频率以及海底信息，还可以计算声线携带声波的强度和相位，这样在经典射线理论的基础上可以简明的描述声场分布，给水声通信时变特性分析和通信系统的设计奠定一定基础。     

MTF measurement method for medical displays by using a bar‐pattern image

Abstract— A modulation‐transfer‐function (MTF) measurement method that uses a bar‐pattern image for medical displays such as liquid‐crystal displays (LCDs) and cathode‐ray tubes (CRTs) has been investigated. A specific bar‐pattern image on the display was acquired with a high‐resolution single‐lens reflex‐type digital camera equipped with a close‐up lens. The MTF was calculated from the amplitudes of the fundamental‐frequency components, which were extracted from the profile data across the bar patterns by using Fourier analysis. Actual comparisons with the conventional line technique were performed for a medical CRT. The adequate accuracy and excellent reproducibility of the method were confirmed. Furthermore, unlike the line method, an advantageous feature which can use an input signal with sufficient amplitude was theoretically proved. Horizontal and vertical MTFs at the central position of the display area were measured up to the Nyquist frequency for several medical displays. From these measurements, this method has the capability to detect slight differences between the displays measured. This proposed method is useful in understanding and quantifying the medical display's performance due to excellent reproducibility and accuracy.