Blending surface generation using a fast and accurate analytical solution of a fourth-order PDE with three shape control parameters

In this paper, we propose to use a fourth-order partial differential equation (PDE) to solve a class of surface-blending problems. This equation has three vector-valued shape control parameters. It incorporates all the previously published forms of fourth-order PDEs for surface blending and can generate a larger class of blending surfaces than existing equations. To apply the proposed PDE to the solution of various blending problems, we have developed a fast and accurate resolution method. Our method modifies Naviers solution for the elastic bending deformation of thin plates by making it satisfy the boundary conditions exactly. A comparison between our method, the closed-form solution method, and other existing analytical methods indicates that the developed method is able to generate blending surfaces almost as quickly and accurately as the closed-form solution method, far more efficiently and accurately than the numerical methods and other existing analytical methods. Having investigated the effects that the vector-valued shape parameters and the force function of the proposed equation have on the blending surface, we have found that they have a significant influence on its shape. They provide flexible user handles that surface designers can use to adjust the blending surface to acquire the desired shape. The developed method was employed in the investigation of surface-blending problems where the primary surfaces were expressed in parametric, implicit, and explicit forms.     

Generating Blending Surfaces with a Pseudo-Lévy Series Solution to Fourth Order Partial Differential Equations

In our previous work, a more general fourth order partial differential equation (PDE) with three vector-valued parameters was introduced. This equation is able to generate a superset of the blending surfaces of those produced by other existing fourth order PDEs found in the literature. Since it is usually more difficult to solve PDEs analytically than numerically, many references are only concerned with numerical solutions, which unfortunately are often inefficient. In this paper, we have developed a fast and accurate resolution method, the pseudo-Lévy series method. Due to its analytical nature, the comparison with other existing methods indicates that the developed method can generate blending surfaces almost as quickly and accurately as the closed form resolution method, and has higher computational accuracy and efficiency than existing Fourier series and pseudo-spectral methods as well as other numerical methods. In addition, it can be used to solve complex surface blending problems which cannot be tackled by the closed form resolution method. To demonstrate the potential of this new method we have applied it to various surface blending problems, including the generation of the blending surface between parametric primary surfaces, general second and higher degree surfaces, and surfaces defined by explicit equations.     

Analytical C smooth blending surfaces

Two factors are important in the generation of blending surfaces for interactive graphical and CAD applications, computational speed and the degree of smoothness. Most surface-blending methods blend surfaces with tangent continuity. However, curvature continuity has recently become increasingly important in various applications. In this paper, we present a method that is able to achieve curvature continuity based on the use of partial differential equations (PDE). The blending surfaces are generated as the solution to a sixth-order PDE with one vector-valued parameter. To achieve interactive performance, we propose an effective analytical method for the resolution of this sixth-order PDE.     

基于物理的能量曲面造型方法构造过渡曲面

讨论了过渡曲面的生成问题,指出了半径过渡和PDE方法构造过渡面的局限性,提出了用基于物理的能量曲面造型方法构造过渡曲面的方法,特别是用于解决管状封闭非周期性曲面和其他曲面间的过渡问题。该文详细讨论了基于物理的能量曲面造型方法构造过渡曲面的原理及求解方法,并给出了飞行器的翼身过渡和三通过渡的实例。     

基于动态偏微分方程构造过渡面

为了实现交互式的偏微分方程曲面造型,针对传统静态偏微分方程构造过渡面存在的不足,提出了基于动态偏微分方程构造C1连续的过渡面,并引入迭代有限差分法求解偏微分方程的数值解,在此基础上构造了光滑过渡曲面.讨论了形状控制因子、密度、阻尼系数等物理参数的变化对曲面形状的影响,其中形状控制因子对生成曲面的形状影响最为明显.造型实例表明,利用动态偏微分方程构造过渡面具有更高的灵活性,大大提高了工业几何设计的交互性,在CAD/CAM中具有重要的应用价值.     

基于结式方法的代数曲面拼接

以同伦连续映射理论为基础,构造代数曲面拼接应该满足的代数方程组。然后,利用结式方法消去相关变元得到拼接曲面方程。两代数曲面拼接时,方程组是两个关于单位区间变元的方程。利用Sylvester结式消去该变元即可得到曲面拼接方程。对于多代数曲面,拼接过程可以考虑为不同种的连续映射。由此得到三种不同的曲面拼接方法,即串接法、过渡法和提升法。串接法可得到较低次的拼接曲面,但适用于代数曲面两两拼接且过渡曲面不相交的情况;过渡法适用于所有情况,但得到拼接曲面比较复杂;提升法是一种较好的算法,拼接时逐个将代数曲面并入拼接曲面中。该算法既可得到最低次拼接方程又适用于一般情况。上述方法的优点是无需考虑代数曲面方程中的变元,仅考虑对新增单位区间变元的处理。因此,算法的计算量小,并且能够预先得到拼接曲面时的计算量。     

Multiple Facial Image Editing Using Edge–Aware PDE Learning

This paper introduces a novel facial editing tool, called edge‐aware mask, to achieve multiple photo‐realistic rendering effects in a unified framework. The edge‐aware masks facilitate three basic operations for adaptive facial editing, including region selection, edit setting and region blending. Inspired by the state‐of‐the‐art edit propagation and partial differential equation (PDE) learning method, we propose an adaptive PDE model with facial priors for masks generation through edge‐aware diffusion. The edge‐aware masks can automatically fit the complex region boundary with great accuracy and produce smooth transition between different regions, which significantly improves the visual consistence of face editing and reduce the human intervention. Then, a unified and flexible facial editing framework is constructed, which consists of layer decomposition, edge‐aware masks generation, and layer/mask composition. The combinations of multiple facial layers and edge‐aware masks can achieve various facial effects simultaneously, including face enhancement, relighting, makeup and face blending etc. Qualitative and quantitative evaluations were performed using different datasets for different facial editing tasks. Experiments demonstrate the effectiveness and flexibility of our methods, and the comparisons with the previous methods indicate that improved results are obtained using the combination of multiple edge‐aware masks.     

Free-form geometric modeling by integrating parametric and implicit PDEs

Parametric PDE techniques, which use partial differential equations (PDEs) defined over a 2D or 3D parametric domain to model graphical objects and processes, can unify geometric attributes and functional constraints of the models. PDEs can also model implicit shapes defined by level sets of scalar intensity fields. In this paper, we present an approach that integrates parametric and implicit trivariate PDEs to define geometric solid models containing both geometric information and intensity distribution subject to flexible boundary conditions. The integrated formulation of second-order or fourth-order elliptic PDEs permits designers to manipulate PDE objects of complex geometry and/or arbitrary topology through direct sculpting and free-form modeling. We developed a PDE-based geometric modeling system for shape design and manipulation of PDE objects. The integration of implicit PDEs with parametric geometry offers more general and arbitrary shape blending and free-form modeling for objects with intensity attributes than pure geometric models     

Corner blending of free-form N-sided holes

Geometric modeling requires constructing blends between surfaces to meet manufacturing specifications, reduce stress concentrations in designs, and enhance aesthetics. Industrial engineers may design parts using different surface types to satisfy design requirements. Surface blending integrates the diverse representations. Because most CAD/CAM software uses parametric representations for curves and surfaces, blending techniques for parametric surfaces are more urgently needed than methods for implicit surfaces. The authors discuss a new method for parametric surface blending. They apply their corner blending technique to three- to six-sided holes and discuss corner blending of three sided holes for three different cases, each presenting different blending challenges     

高质量快速全景场景再现技术

目的 色彩纠正和图像融合是生成高质量全景场景图像的关键技术。色彩纠正中参考图像的选择以及图像融合算法,决定着所生成全景图像的质量和速度。现有方法在确定一幅图像是否适合作为参考图像时,需要遍历所有其他图像,计算其作为参考图像进行色彩纠正的效果,复杂度高,速度慢;在图像融合时存在融合质量与融合速度之间的矛盾。因此,如何快速生成高质量的全景图像就成为全景场景再现的主要诉求。为此本文提出优化的参考图像自动选择的色彩纠正方法和基于重叠区域划分的分区融合方法。方法 针对参考图像选择算法复杂度高的问题,根据图像质量与稳定性通常呈反比关系的事实,采用贪婪策略,选择质量最差的图像在色彩纠正前后的相似度,作为是否选择当前图像作为参考图像的评价指标,在保证参考图像满足色彩纠正需求的前提下,大幅降低计算复杂度。针对融合质量与融合速度之间的矛盾,提出分区融合：将重叠区域划分为接缝区域和非接缝区域,利用泊松融合的接缝不可见性和线性融合实现速度快的特性分别对接缝区域和非接缝区域进行融合,既保证融合的质量,又加快融合速度。在此基础上,加入简单点光源,解决上述过程产生的光线一致性问题,进一步提高图像质量。结果 采用主观和客观相结合的方法对结果进行评估。主观方面,本文算法生成的全景图像色彩基本实现平滑过渡且图像原始信息保留完整。客观方面,色彩纠正前后图像的结构相似度（SSIM）控制在0.850.99之间,时间复杂度由原来的O（n²）降低到O（n）;分区融合后图像信息熵接近于泊松融合,但时间消耗降低72%。采用基于PC端的问卷调查法和OG-IQA算法将本文算法与PTGui、OpenCV、Xiong方法生成的全景图质量进行对比,在大多数情况下本文算法均优于上述算法。结论 实验表明,本文算法适用于多种场景,在保证目视效果良好的前提下,时间消耗降低,可广泛应用于医学、数字旅游、遥感等领域。     

A survey of partial differential equations in geometric design

Computer-aided geometric design is an area where the improvement of surface generation techniques is an everlasting demand, since faster and more accurate geometric models are required. Traditional methods for generating surfaces were initially mainly based upon interpolation algorithms. Recently, partial differential equations (PDE) were introduced as a valuable tool for geometric modelling, since they offer a number of features from which these areas can benefit. This work summarizes the uses given to PDE surfaces as a surface generation technique together with some other applications to computer graphics.     

基于OpenGL的复杂曲面的纹理映射

介绍利用OpenGL和Visual C 6.0进行复杂曲面的纹理映射。利用解二次随圆偏微分方程的方法，得到任意曲面到平面的共形映射。此方法可以自动分配纹理坐标到复杂的没有起伏的曲面，有效地克服复杂曲面的自动纹理映射的变形，避免了纹理扰动。     

Dynamic PDE-based surface design using geometric and physical constraints

PDE surfaces, which are defined as solutions of partial differential equations (PDEs), offer many modeling advantages in surface blending, free-form surface modeling, and specifying surface’s aesthetic or functional requirements. Despite the earlier advances of PDE surfaces, previous PDE-based techniques exhibit certain difficulties such as lack of interactive sculpting capabilities and restrained topological structure of modeled objects. This paper presents an integrated approach that can incorporate PDE surfaces into the powerful physics-based modeling framework, to realize the full potential of PDE methodology. We have developed a prototype system that allows interactive design of flexible topological surfaces as PDE surfaces and displacements using generalized boundary conditions as well as a variety of geometric and physical constraints, hence supporting various interactive techniques beyond the conventional boundary control. The system offers a set of sculpting toolkits that allow users to interactively modify arbitrary points, curve spans, and/or regions of interest across the entire PDE surfaces and displacements in an intuitive and physically meaningful way. To achieve real-time performance, we employ several simple, yet efficient numerical techniques, including the finite-difference discretization, the multigrid-like subdivision, and the mass-spring approximation of elastic PDE surfaces and displacements. In addition, we present the standard bivariant B-spline finite element approximations of dynamic PDEs, which can subsequently be sculpted and deformed directly in real-time subject to the intrinsic PDE constraints. Our experiments demonstrate many attractive advantages of the physics-based PDE formulation such as intuitive control, real-time feedback, and usability to both professional and common users.     

基于变分法和偏微分方程的图像彩色化方法

针对传统的图像彩色化算法易产生的图像边缘颜色模糊等问题,提出一种基于变分法和偏微分方程的图像彩色化算法,结合整体变分(TV)模型和金兹堡一朗道(GL)模型的特点,通过求泛函极值建立偏微分方程,使图像的光滑区域和边界区域都能有效的实现颜色扩散,对偏微分方程进行数值求解得到彩色化结果.实验结果表明,与其它偏微分方程彩色化技术相比,该方法能得到清晰自然的彩色化图像和较高的图像质量.     

边界满足C0或C1连续条件的光顺过渡面的构造

针对联结两个曲面片的过渡曲面构造问题,使用偏微分方程中的热传导方程,详细研究了如何构造边界满足C⁰或C¹连续条件的光顺过渡面的偏微分方程模型,讨论了初始曲面及方程参数对光顺过渡面的影响。     

加权型曲率保持 PDE 图像滤波方法

提出了一种加权型曲率保持偏微分方程(Partial differential equation, PDE)滤波方法.传统曲率保持PDE 滤波方法未考虑各积分曲线可能经历不同的图像结构,如此影响了其对图像边缘的保持能力.在此基础上, 利用局部图像方向信息为不同积分曲线设计了相应的权重,得到了一种张量驱动的加权型曲率保持PDE滤波方法. 实验结果表明该方法在滤波的同时能较好地保持图像中边缘与曲率结构,且对图像具有一定增强能力.     

Reversible data hiding based on PDE predictor

In this paper, we propose a prediction-error expansion based reversible data hiding by using a new predictor based on partial differential equation (PDE). For a given pixel, PDE predictor uses the mean of its four nearest neighboring pixels as initial prediction, and then iteratively updates the prediction until the value goes stable. Specifically, for each pixel, by calculating the gradients of four directions, the direction with small magnitude of gradient will be weighted larger in the iteration process, and finally a more accurate prediction can be obtained. Since PDE predictor can better exploit image redundancy, the proposed method introduces less distortion for embedding the same payload. Experimental results show that our method outperforms some state-of-the-art methods.     

基于激光诊断技术的脉冲爆震发动机多参数自动测试系统

脉冲爆震发动机(以下简称PDE)的燃烧产物具有高温、高压、高污染、极快的脉动速度等特点,为了实时检测PDE工作过程中各参数的变化,改进PDE的设计性能.应用激光诊断技术,虚拟仪器技术,研制了PDE多参数自动测试系统(以下简称系统).叙述了该系统的工作原理、系统硬件结构和软件的设计方法.运用LabWindows/CVI与MATLAB混合编程提高了系统的开发效率,使系统功能更加灵活、强大.用该系统对一PDE实验样机作了实时检测,得到了PDE工作过程中的燃烧产物的温度及组分浓度.结果表明:该系统的时间分辩率小于1 ns,获取的CO2、CO、及HC的值反映了PDE的燃烧效率.这是传统的测量方法无法实现的.     

Max-plus eigenvector representations for solution of nonlinear H/sub /spl infin// problems: basic concepts

The H/sub /spl infin// problem for a nonlinear system is considered. The corresponding dynamic programming equation is a fully nonlinear, first-order, steady-state partial differential equation (PDE), possessing a term which is quadratic in the gradient. The solutions are typically nonsmooth, and further, there is nonuniqueness among the class of viscosity solutions. In the case where one tests a feedback control to see if it yields an H/sub /spl infin// controller, the PDE is a Hamilton-Jacobi-Bellman equation. In the case where the "optimal" feedback control is being determined as well, the problem takes the form of a differential game, and the PDE is, in general, an Isaacs equation. The computation of the solution of a nonlinear, steady-state, first-order PDE is typically quite difficult. In this paper, we develop an entirely new class of methods for obtaining the "correct" solution of such PDEs. These methods are based on the linearity of the associated semigroup over the max-plus (or, in some cases, min-plus) algebra. In particular, solution of the PDE is reduced to solution of a max-plus (or min-plus) eigenvector problem for known unique eigenvalue 0 (the max-plus multiplicative identity). It is demonstrated that the eigenvector is unique, and that the power method converges to it. An example is included.