共形几何代数--几何代数的新理论和计算框架

共形几何代数是一个新的几何表示和计算工具.作为几何的高级不变量和协变量系统的结合,它为经典几何提供了统一和简洁的齐性代数框架,以及高效的展开、消元和化简算法,从而可以进行极其复杂的符号几何计算,在几何建模与计算方面表现出很大的优势.主要讲述共形几何代数的产生背景和意义,共形几何代数的数学理论和它最有特色的几个部分,包括Grassmann结构、统一几何表示和旋量作用、基本不变量系统和高级不变量系统、新的计算思想、展开和化简技术等.     

几何代数的高阶逻辑形式化

几何代数是一种用于描述和计算几何问题的代数语言.由于它统一的表达分析和不依赖于坐标的几何计算等优点现已成为数学分析、理论物理、几何学、工程应用等领域重要的理论基础和计算工具, 然而利用几何代数进行计算和建模分析的传统方法,如数值计算方法和符号方法等都存在计算不精确或者不完备等问题.高阶逻辑定理证明是验证系统正确的一种严密的形式化方法.本文在高阶逻辑证明工具HOL-Light中建立了几何代数系统的形式化模型,主要包括片积、多重矢量、外积、内积、几何积、几何逆、对偶、基矢量运算和变换算子等的形式化定义和相关性质定理的证明.最后为了说明几何代数形式化的有效性和实用性,本文在共形几何代数空间中对刚体运动问题提供了一种新的简单有效的形式化建模与验证方法.     

共形几何代数与几何不变量的代数运算

几何不变量的使用是计算机视觉和图形学的一个重要手段.发现一个不变量后,如何找到它与其他不变量的关系,是实际应用中的一个重要问题,这种关系的探讨主要依靠在不变量层次上的代数运算.文中介绍了共形几何代数中的基本、高级和有理不变量如何在几何问题中自然出现,它们之间如何进行代数运算,以及如何通过不变量的化简,自然地得到几何条件的充分必要化和几何定理的完全化.几何定理的机器证明作为几何定理完全化的副产品,被发展成几何定理的关系定量化,这种量化的几何还原就是几何定理的自然推广.几何不变量之间的几何关系的计算是这些技术的一个具体应用.     

基于CGA理论的虹膜定位算法的研究*

为了能够有效提高虹膜检测和定位的质量、准确性和速度,排除光线照射、噪声或是拍摄的角度等不利的因素对定位结果的影响,本文提出一种虹膜的定位算法。该方法首先在对图像进行预处理的基础之上,运用共形几何代数理论的思想,将欧式空间当中的几何量变换到共形几何代数空间当中去,这些几何量在共形几何代数空间当中都是以统一矢量的形式表示,从而这些几何量之间的计算更加的方便、简洁,最后借助Radon变换对目标圆,即虹膜的内外圆所在的平面进行检测,实现虹膜的定位。实验结果表明,该方法简单快速且虹膜定位结果准确,抗干扰的能力强,最重要的是实现了对虹膜内外边缘的同时定位,这对提高虹膜检测和定位的效率有着重要的意义。     

图学与几何

本文讨论图学与几何的关系。从形是图之源、图形/图像的本质是几何这个基本认 识出发,指出图的产生、表示、处理与传播过程都是在处理几何的定义、变换与其间关系,因 此图与图学的基础是几何,图学计算的基础是几何计算。基于图学与计算的内涵分析,剖析了 图形计算的本质、矛盾和关键技术。讨论了代数与几何在图学计算中各自的作用与利弊,强调 人在算法设计中的主导地位。指出生成一幅图或构建一个模型的主要工作不是决定构成该图或 模型的元素本身,而在于找出元素之间的相互关系。针对图形图像已经成为计算的主要对象与 结果表述,介绍了一种基于几何的形计算机制,弥补常规数计算的不足,追求“形思考、数计算” 的几何计算新模式。     

一种基于几何的形计算机制

提出一种几何问题几何化的形计算机制。它综合了几何、代数、画法几何及现代计 算工具等理论、方法与技术,实现“三维思维,二维图解,一维计算”多维空间的融合。从更宏 观的几何角度构筑算法框架,是对常规数计算的补充,可用于相当宽泛的一类几何计算。     

点至平面代数曲线正交投影计算的混合算法EI北大核心CSCD

点至平面代数曲线的正交投影计算在计算机图形学、计算机辅助几何设计领域,特别是交互式设计等应用中有着非常重要而广泛的运用.基于牛顿梯度下降法、切线和曲率圆形成中点的中点脚点法,以及混合几何加速正交法的计算方法,提出一种混合算法用于计算点到平面代数曲线的正交投影问题.首先,采用牛顿梯度下降法使初始迭代点落在平面代数曲线上;其次,利用切线和曲率圆所形成的中点作为脚点,再结合牛顿梯度下降法,将落在平面代数曲线上的迭代点逐渐挪动至正交投影点很靠近位置;最后,使用混合几何加速正交法得到正交投影点.采用3个封闭平面代数曲线实例进行实验,通过收敛性计算验证,结果表明当测试点比较远或代数曲线次数比较高时,该算法是鲁棒和高效的.     

基于几何代数的散焦模糊高维图像恢复

几何代数易于对高维空间几何进行计算和分析,应用几何代数的这一特性,将彩色图像表示为高维几何空间中的点元素,利用几何代数描述图像的变换关系,将图像的散焦变换看作是高维空间中点元素的平移运动。通过分析模糊图像以及其衍生出的相关模糊图像对应在高维几何空间中点之间的分布关系的研究,计算出空间中复原图像的点分布位置。实验结果验证了该方法的有效性。     

图学计算基础

首先探索图学计算的内涵,剖析图形计算的矛盾,找出图形计算的关键, 构造图学的公共基础,指出图的本质不是决定于构成图的图元本身,而决定于图元之间的相 互关系。其次,分析了图形计算的最新研究进展,总结出几何化方法的优势及其基础理论研 究的迫切性。进而提出一个基于“几何问题几何化”的几何计算理论体系与实施框架。阐述 了以几何为思路,在构筑稳定的图学计算平台方面的最新原创理论与机制,包括从形的角度 去统一、研究、发展图学计算基础的理论、形计算的机制等。     

空间两三角形的相交问题

重点考虑几何奇异问题,同时兼顾算法的效率。运用“分而治之”的方法从 一维解得到二维解,进而得到三维解,将空间问题变为平面问题、线性问题。基于几何代数 化依赖于坐标系,引入“计算坐标系”,简化了几何的表述与关系的类型,使“几何奇异”状态 最后归结为平面上线段被三角形裁剪时的共点、共线问题,简单而明晰,从而可从理论上保 证算法的鲁棒性,以平面处理的形式给出了两个空间三角形求交的完整解决方案。测试证明, 几何关系、几何奇异类型与计算的简化足以弥补因“变换”而增加的额外开销。算法的速度也 能达到实用要求——在笔记本电脑上也能达到每秒100 万对三角形的相交计算。     

共形几何代数与运动和形状的刻画

共形几何代数在基于运动和形状刻画的视觉和图形学若干问题中的应用,反映了它能够提供统一和有效的表示和算法,这些应用主要集中在采纳几何体的Grassmann分级表示以及刚体运动的旋量和扭量表示.着重介绍了Grassmann分级表示如何被应用于单眼视觉问题并带来解决方法的简化;通过对刚体运动不同表示的分析,介绍旋量和扭量表示如何克服刚体运动蹬矩阵表示中参数空间具有过多非线性约束的缺点,从而为姿态估计、形状逼近和曲线拼接等问题的解决提供简化方案.     

A Geometric Approach for the Theory and Applications of 3D Projective Invariants

A central task of computer vision is to automatically recognize objects in real-world scenes. The parameters defining image and object spaces can vary due to lighting conditions, camera calibration and viewing position. It is therefore desirable to look for geometric properties of the object which remain invariant under such changes in the observation parameters. The study of such geometric invariance is a field of active research. This paper presents the theory and computation of projective invariants formed from points and lines using the geometric algebra framework. This work shows that geometric algebra is a very elegant language for expressing projective invariants using n views. The paper compares projective invariants involving two and three cameras using simulated and real images. Illustrations of the application of such projective invariants in visual guided grasping, camera self-localization and reconstruction of shape and motion complement the experimental part.     

Omnidirectional Robot Vision Using Conformal Geometric Computing

In this paper, we show how to use the conformal geometric algebra (CGA) as a framework to model the different catadioptric systems using the unified model (UM). This framework is well suited since it can not only represent points, lines and planes, but also point pairs, circles and spheres (geometric objects needed in the UM). We define our model using the great expressive capabilities of the CGA in a more general and simpler way, which allows an easier implementation in more complex applications. On the other hand, we also show how to recover the projective invariants from a catadioptric image using the inverse projection of the UM. Finally, we present applications in navigation and object recognition. Carlos Alberto López-Franco is a doctoral student at CINVESTAV, GEOVIS Laboratory, Unidad Guadalajara, México. He received in 2003 the M.S. degree in Computer Science from CINVESTAV, Unidad Guadalajara. His scientific interests are in the fields of computer vision, robotics and the applications of geometric algebra for mobile robots. Eduardo Jose Bayro-Corrochano gained his Ph.D. in Cognitive Computer Science in 1993 from the University of Wales at Cardiff. From 1995 to 1999 he has been Researcher and Lecturer at the Institute for Computer Science, Christian Albrechts University, Kiel, Germany, working on applications of geometric Clifford algebra to cognitive systems. At present is a full professor at CINVESTAV Unidad Guadalajara, México, Department of Electrical Engineering and Computer Science. His current research interest focuses on geometric methods for artificial perception and action systems. It includes geometric neural networks, visually guided robotics, color image processing, Lie bivector algebras for early vision and robot maneuvering. He developed the quaternion wavelet transform for quaternion multi-resolution analysis using the phase concept. He is associate editor of Robotics and Journal of Advanced Robotic Systems and member of the editorial board of Journal of Pattern Recognition, Journal of Mathematical Imaging and Vision, Iberoamerican Journal of Computer and Systems and Journal Of Theoretical And Numerical Approximation. He is editor and author of the following books: Geometric Computing for Perception Action Systems, E. Bayro-Corrochano, Springer Verlag, 2001; Geometric Algebra with Applications in Science and Engineering, E. Bayro-Corrochano and G. Sobczyk (Eds.), Birkahauser 2001; Handbook of Geometric Computing for Pattern Recognition, Computer Vision, Neurocomputing and Robotics, E. Bayro-Corrochano, Springer Verlag, 2005. He has published over 120 refereed journal, book chapters and conference papers.     

Motor Algebra for 3D Kinematics: The Case of the Hand-Eye Calibration

In this paper we apply the Clifford geometric algebra for solving problems of visually guided robotics. In particular, using the algebra of motors we model the 3D rigid motion transformation of points, lines and planes useful for computer vision and robotics. The effectiveness of the Clifford algebra representation is illustrated by the example of the hand-eye calibration. It is shown that the problem of the hand-eye calibration is equivalent to the estimation of motion of lines. The authors developed a new linear algorithm which estimates simultaneously translation and rotation as components of rigid motion.     

New Geometric Methods for Computer Vision: An Application to Structure and Motion Estimation

We discuss a coordinate-free approach to the geometry of computer vision problems. The technique we use to analyse the three-dimensional transformations involved will be that of geometric algebra: a framework based on the algebras of Clifford and Grassmann. This is not a system designed specifically for the task in hand, but rather a framework for all mathematical physics. Central to the power of this approach is the way in which the formalism deals with rotations; for example, if we have two arbitrary sets of vectors, known to be related via a 3D rotation, the rotation is easily recoverable if the vectors are given. Extracting the rotation by conventional means is not as straightforward. The calculus associated with geometric algebra is particularly powerful, enabling one, in a very natural way, to take derivatives with respect to any multivector (general element of the algebra). What this means in practice is that we can minimize with respect to rotors representing rotations, vectors representing translations, or any other relevant geometric quantity. This has important implications for many of the least-squares problems in computer vision where one attempts to find optimal rotations, translations etc., given observed vector quantities. We will illustrate this by analysing the problem of estimating motion from a pair of images, looking particularly at the more difficult case in which we have available only 2D information and no information on range. While this problem has already been much discussed in the literature, we believe the present formulation to be the only one in which least-squares estimates of the motion and structure are derived simultaneously using analytic derivatives.     

基于运动矢量的抗几何攻击视频水印方案

许多数字图像和视频水印方案都无法抵挡几何变换攻击,即使是轻微的旋转、缩放、平移也将会破坏水印的同步,这就使得水印难以检测出来。为解决此难题,提出了一种新颖的视频水印算法。该算法利用了运动矢量特征的相对关系不易随几何失真变换而改变的特性,使得水印具有内在的抗几何失真的鲁棒性。该算法中,水印嵌入和检测时,不仅可根据平均运动矢量的方向和幅度确定水印模板的位置和方向,而且水印检测不需要原始视频。实验表明,该水印算法对几何变换攻击具有较强的鲁棒性。