多维几何

Events,Operation and Relation of Sets, Classical Probability, Geometrical Probability , Statistical Stability of a Frequency, Axioms of Probability, Conditional Probability, Total Probability Theorem, Bayes' Rule,Independent Events,Independent Repeated Trials, One Dimensional Random Variables, Discrete Random Variables, Distribution Function of a Random Variables , Continuous Random Variables, Normal Distribution, Distribution of a Function of a Random Variable, Multidimensional Random Variables, Joint Distribution Function, Marginal Distribution Function,Discrete Two—Dimensional Random Variables,Continuous Two—Dimensional Random Variables, Independent Random Variables, Distribution of Functions of Random Variables,Expectation,Variance, Covariance, Coefficient of Correlation, Bivariate Normal Distribution, Law of Large Numbers, The Central Limit Theorems, Sample and Population ,Chi—Squared, T and F Distributions , Sampling Distributions , Point Estimation , Interval Estimation , Testing Hypotheses , A Test of Significance for Parameters in a Single Sample From a Normally Distributed Population , A Test of Significance for Parameters in Two Sample From Normally Distributed Populations .

本课程的主要内容：概率的概念与运算、随机变量及其分布、随机变量的数字特征与极限定理、数理统计的基本概念、估计和检验的基本方法，随机事件与概率随机事件、事件的关系与运算、几何概率、统计概率等，条件概率、全概率公式、贝叶斯公式、事件的独立性、二项概率公式，随机变量的概念、离散型随机变量、随机变量的分布函数、连续型随机变量、随机变量函数的分布，多维随机变量及其分布函数、边缘分布函数、随机变量的独立性、二维随机变量函数的分布，数学期望、方差、协方差和相关系数、大数定律、中心极限定理，总体与样本， X 2-分布、 t-分布和 F-分布，统计量及抽样分布，假设检验的基本概念、单个正态总体参数的显著性检验、两个正态总体参数的显著性检验。

The error source of CMM is complex, and the current main research is its geometric error. In the paper, 21 items of geometric error are measured accurately, simply and fast with the 2-D laser interferometer and four quadrants photo detector. The measurement gets rid of measuring so many lines and several diagonal lines in linear displacement method, and also solves the question that error measure of large-scale CMM can not be measured with 1-D ball array and ball plate conveniently.

坐标测量机的误差源纷繁复杂，当前主要研究的是其几何误差，课题中研究了一种采用二维激光干涉仪和四象限光电探测器相结合的测量方法，对三坐标测量机的21项几何误差进行了准确、简便、快速的测量，摆脱了线位移法测量直线过多且包含多条对角线测量的弊端，同时也解决了一维球列和球板不方便检测的大型测量机的误差检测问题。

Whether in its narrow or broad definition, stereology is essentially an interdisciplinary branch of science about 2D and 3D structures and images,especially the three-dimensional interpretation of flat images by criteria of geometric probability.

不同定义中有一点是共同的，即体视学是一门关于多维几何结构及图像的边缘和交叉科学分支，是不同学科中获得三维显微组织结构几何形态信息的基本工具。

Based on the relationship between the structure and function, a lot of research have been performed on predicting base-pair which is indispensable to the RNA three-dimensional strcutrue. The tools and methods in Combinatorics and Statistics will play important roles in studying linear sequences of biomolocular units . Similarly, we can present the geometric representation of biological sequences and structures. So the genometric topology and group theory are important also.

根据RNA的结构与功能的关系，在预测碱基配对的类型方面进行了相当多的研究，这也是确定RNA三维构象的必不可少的一部分，组合学和统计学的工具和方法在研究生物序列和生物结构上起到了很大的作用，同样我们可以用几何方法来表示生物序列和二级结构，几何拓扑和群论也起到了很大的作用。

Last, this paper introduced some concepts of multiscale geometrical analysis and curvelet transform theory. Because curvelet can showed this feature more 'sparsely' in the function included singular curve or singular side in two dimension space or multi-dimension space. This paper provided a robust digital watermarking technology algorithm based on curvelet transform. watermarking information was embedded into coarse scale coefficients after curvelet transforming, choose right embedding strength. Testing proved it has good invisibility and robustness.

最后，介绍了多尺度几何分析和曲波变换理论的相关概念，鉴于曲波能对二维乃至高维空间中含奇异曲线或者曲面的函数进行更"稀疏"地表示这一特性，本章提出了一种基于Curvelet变换的鲁棒数字图像水印算法，该算法将水印信息嵌入到经曲波变换后的图像的粗尺度系数中，合理选择的嵌入强度，试验表明该算法具有较好的隐蔽性和鲁棒性。

The multi-dimensional signals measured by sensors with a machine working under normal condition or fault condition, can be used to build Q statistic and T^2 statistic respectively to detect the state of the machine. Meanwhile, the feature vector is constructed by the Q value and the T^2 value, the geometrical distance between the normal feature vector and the feature vector H when the machine goes wrong is used to realize the fault location. The results show that the method is feasible.

利用多个传感器测量的异步电机多维信号参量，构建电机在正常工作和发生故障时的Q统计和T^2统计，以实现电机的状态检测；利用Q统计和T^2统计值构建电机的状态特征向量，通过比较度量当前电机的特征向量与电机发生故障时的特征向量H的几何距离来实现电机故障的定位与分离实验证明，该方法可以有效地实现故障的诊断与分离。

By time-varying filtering, we realize the signal selectivity and interference suppression, especially for the signals with time-varying power spectrum. It is shown herein that conventional eigenstructure based methods are lower dimensional special case of the time-frequency subspace fitting method. The closed-form array response estimation algorithms for blind source separation in wireless communication is also developed in the dissertation by exploiting the data structure of second-order statistics of the array output in the presence of multipath. The proposed method achieves array response estimation with little constraint on signal property and propagation environment such as scatters or angular spread. Moreover, the array utilized here can be of arbitrary geometry and even uncalibrated. Since the closed-form solution is obtained, no misconvergency may occur and large amount of computation can be saved. The equivalent feed model and corresponding optimization designation are proposed to realize the downlink multi-user selective transmission. By coherent and non-coherent superposition of the electromagnetical field, multiple beams can be formed using a single array, so that each co-channel user signal is transmitted in each beam and will not interfered with each other. The proposed system scheme implements the digital beamforming and the channel allocation in the base band, therefore is suitable for the utilization of the radio software technology. The feasibility and effectiveness of the proposed model are proved by the RF experiment.

在窄带条件下，GSC－SSF方法还可以提高阵列的虚拟孔径，从而提高估计的性能；可以证明，Cyclic DOA 和SC-SSF方法可看作该方法的特例或近似； 3、提出了二维GSC－SSF方法和宽带谱相关DOA矩阵方法，利用信号的循环平稳特性解决了宽带条件下二维 DOA 估计问题，并具有阵元数少、计算量较小的特点； 4、提出了时频子空间拟合和宽带时频DOA估计算法，将联合时频分析的理论和方法引入到阵列信号的空间谱估计研究中，利用时变滤波实现信号选择性测向和干扰抑制，特别适合于谱时变信号；可以证明传统特征结构法可看作是时频子空间拟合方法的特例； 5、提出了多用户环境下阵列响应的估计方法——CARE方法，该方法对信号结构和传播参数（主要是散射源分布与角度散布）的约束较少，适用于各种几何结构的阵列甚至未校准的阵列，能够直接得到阵列响应的闭式解，因此计算量小、稳健性好，不存在收敛问题； 6、提出了针对多用户下行选择性发送的阵列等效激励模型、优化设计方法及其系统实现方案，通过空间场的相干叠加与非相干叠加使得单一阵列能够形成多个波束，并保证各波束内的用户信号各不相同且相互干扰最小；其系统方案具有在基带实现数字波束成形、信道分配并便于软件无线电实现的特点，其有效性已得到外场实测数据的证实。

The algorithm is applied in the segmentation of Magnetic Resonance Imaging of human brain for the 3D reconstruction on dependent of different human tissue structures, in which the problem on wraps between different tissues and fuzzy edges between the different regions in medical volume data may better be settled, which can not be well done by the typical"hard"segmentation using binary logical representation.(3) An accelerated algorithm of volume rendering with X-ray perspective and Phong shading effect based on textures mapping and blending is presented for realizing 3D reconstruction of medical volume data by 2D texture mapping and blending, in which the accelerated algorithm of window transformation is utilized to computing voxels' gray scales and opacities to produce X-ray perspective images, and Phone illumination model is used to import world light sources and an unit spherical surface under the same shading environment as the actual volume rendering is employed to simulate the real reflectional light of per-voxel so that 3D reconstruction possesses illumination effect in the world. Furthermore, 3D matrix transformations of degree rotation and other operations of 3D object are studied, and a fast equivalent method of corresponding shading correction is brought forward to improve its interaction speed in volume rendering process.

该方法利用体数据的窗口变换加速算法将医学体数据转换为计算机显示值，可以得到具有X光透视效果三维图像；该方法提出的Phong明暗处理加速算法利用Phong光照模型引入外部光源，使用一单位球面体来仿真体数据的法线矢量集所产生的反射光强，以快速得到每一个体素的反射光强度，使物体的三维重建达到现实世界中人眼直接观察的逼真效果；此外，还研究了三维重建物体的三维旋转等矩阵变换，提出了一种Phong明暗修正的快速算法来保证了体绘制中三维几何变换的多视角观察的交互速度。

Through simulation of the real process of thomcolumbar burst fracture and use of modem dynamics techniques, the present study was (1) to investigate the kinetic relationship between structural damage and impact energy absorption and dynamics mechanism of thoracolumbar burst fracture by quantitative analysis of various injured segments;(2) to explore the instability mechanism of L1 vertebral burst fracture and its injury threshold by three-dimensional analysis of the injured segment with stereophotogrammetry in combination of transient physical parameters, anatomy and image;(3) to explore the corresponding relationships between impact energy, geometry parameters and biomechanics by geometry and biomechanical analysis of thoracolumbar burst fracture, which may provide an objective standard for evaluation of spinal injury severity and experimental evidence for adoption of biomechanical treatment in clinic;(4) to evaluate the fixation effect of different instruments and the effect of intervertebral bone graft on segment fixation by analysis of the threedimensional stability of different internal fixation instruments and their decompression of the spinal canal, which may provide experimental evidence for therapeutical selection for thoracolumbar burst fracture in future; and finally to investigate the effect of various instrument fixation on dynamical characteristics by comparison and quantitative analysis of frequency and amplitude responses, which may provide certain theories and experimental evidence for application of vibration test to judge the stability of the spine.

本研究从胸腰段脊柱爆裂骨折发生的实际过程出发，以现代动力学检测及计算机多通道高性能数据采集分析系统等高新技术为基础，实时量化分析损伤节段的工程参数，揭示了结构破坏与能量吸收的变化规律，探讨了胸腰段脊柱爆裂骨折的瞬态损伤机制；利用三维立体摄像技术，对不同损伤程度的节段进行三维运动分析，结合瞬态物理参数、影像学和病理解剖，明确了〓椎体爆裂骨折的失稳机制及损伤阈值；通过对胸腰椎爆裂骨折的几何学变化及生物力学分析，明确了撞击能量与几何参数、几何参数与生物力学的相应关系，为评判脊柱损伤程度提供了客观标准，为治疗中采取适宜的生物力学方法提供了实验依据；通过对比分析不同内固定器械的三维稳定性和对椎管的减压作用，评价不同器械的复位固定作用，同时比较椎体间植骨对节段固定作用的影响，为今后临床胸腰段脊柱爆裂骨折的治疗选择提供依据；利用振动测试与分析技术，比较及量化分析了不同致伤状况的频幅响应特征，比较分析了各种器械固定对动力学特性的影响，为今后临床建立基于振动测试与分析技术判断脊柱稳定性的诊断方法，提供了一定的理论和实验依据。

Considing the complication of crenellated radial labyrinth geometry, a Taylor-Galerkin finite element scheme is developed to solve the multi-dimensional unsteady viscous convection-diffusion equation.

针对锯齿型迷宫密封几何形状的复杂性，发展了求解多维粘性非定常对流扩散方程的Taylor—Galerkin有限元法。

But we don't care about Battlegrounds.

但我们并不在乎沙场中的显露。

Ah! don't mention it, the butcher's shop is a horror.

啊！不用提了。提到肉，真是糟透了。