Mathematical Aspects of SU (2) and SO(3,R) Derived from Two-Mode Realization in Coordinate-Invariant Form

Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and the vector parameter c of Fyodorov are developed. The one-dimensional root scheme of SU (2) is embedded in two-dimensional root schemes of some higher Lie groups, in particular, in inhomogeneous Lie groups and is represented in text and figures. The two-dimensional fundamental representation of SU (2) is calculated and from it the composition law for the product of two transformations and the most important decompositions of general transformations in special ones are derived. Then the transition from representation of SU (2) to of is made where in addition to the parametrization by vector the convenient parametrization by vector c is considered and the connections are established. The measures for invariant integration are derived for and for SU (2) . The relations between 3D-rotations of a unit sphere to fractional linear transformations of a plane by stereographic projection are discussed. All derivations and representations are tried to make in coordinate-invariant way.

Combined Method of Datum Transformation Between Different Coordinate Systems

The similarity transformation model between different coordinate systems is not accurate enough to describe the discrepancy of them.Therefore,the coordinate transformation from the coordinate frame with poor accuracy to that with high accuracy cannot guarantee a high precision of transformation.In this paper,a combined method of similarity transformation and regressive approximating is presented.The local error accumulation and distortion are taken into consideration and the precision of coordinate system is improved by using the recommended method

GEOMETRICALLY INVARIANT WATERMARKING BASED ON RADON TRANSFORMATION

The weakness of classical watermarking methods is the vulnerability to geometrical distortions that widely occur during normal use of the media. In this letter, a new image- watermarking method is presented to resist Rotation, Scale and Translation (RST) attacks. The watermark is embedded into a domain obtained by taking Radon transform of a circular area selected from the original image, and then extracting Two-Dimensional (2-D) Fourier magnitude of the Radon transformed image. Furthermore, to prevent the watermarked image from degrading due to inverse Radon transform, watermark signal is inversely Radon transformed individually. Experimental results demonstrate that the proposed scheme is able to withstand a variety of attacks including common geometric attacks.

Hawking effect and quantum nonthermal radiation of an arbitrarily accelerating charged black hole using a new tortoise coordinate transformation

Using a new tortoise coordinate transformation, this paper investigates the Hawking effect from an arbitrarily accelerating charged black hole by the improved Damour-Ruffini method. After the tortoise coordinate transformation, the Klein-Gordon equation can be written as the standard form at the event horizon. Then extending the outgoing wave from outside to inside of the horizon analytically, the surface gravity and Hawking temperature can be obtained automatically. It is found that the Hawking temperatures of different points on the surface are different. The quantum nonthermal radiation characteristics of a black hole near the event horizon is also discussed by studying the Hamilton-Jacobi equation in curved spacetime and the maximum overlap of the positive and negative energy levels near the event horizon is given. There is a dimensional problem in the standard tortoise coordinate and the present results may be more reasonable.

A complete solution of an improved universal 3D coordinate similarity transformation model

When linearizing three-dimensional(3 D)coordinate similarity transformation model with large rotations,we usually encounter the ill-posed normal matrix which may aggravate the instability of solutions.To alleviate the problem,a series of conversions are contributed to the 3 D coordinate similarity transformation model in this paper.We deduced a complete solution for the 3 D coordinate similarity transformation at any rotation with the nonlinear adjustment methodology,which involves the errors of the common and the non-common points.Furthermore,as the large condition number of the normal matrix resulted in an intractable form,we introduced the bary-centralization technique and a surrogate process for deterministic element of the normal matrix,and proved its benefit for alleviating the condition number.The experimental results show that our approach can obtain the smaller condition number to stabilize the convergence of the interested parameters.Especially,our approach can be implemented for considering the errors of the common and the non-common points,thus the accuracy of the transformed coordinates improves.

Applying invariant eigen-operator method to deriving normal coordinates of general classical Hamiltonian

For classical Hamiltonian with general form H = 1/2∑ijMijpipj＋1/2∑ijLijqiqj we find a new convenient way to obtain its normal coordinates, namely, let H be quantised and then employ the invariant eigen-operator （IEO） method （Fan et al. 2004 Phys. Lett. A 321 75） to derive them. The general matrix equation, which relies on M and L, for obtaining the normal coordinates of H is derived.

Correlation of coordinate transformation parameters

Coordinate transformation parameters between two spatial Cartesian coordinate systems can be solved from the positions of non-colinear corresponding points. Based on the characteristics of translation, rotation and zoom components of the transformation, the complete solution is divided into three steps. Firstly, positional vectors are regulated with respect to the centroid of sets of points in order to separate the translation compo- nents. Secondly, the scale coefficient and rotation matrix are derived from the regulated positions independent- ly and correlations among transformation model parameters are analyzed. It is indicated that this method is applicable to other sets of non-position data to separate the respective attributions for transformation parameters.

Grid Model for High-accuracy Coordinate Transformation of China Geodetic Coordinate System 2000

After implementing CGCS2000,establishing grid models for high-accuracy coordinate transformation which are mainly used to transform border lines and coordinate grids of topographic maps becomes an important issue in mapping applications.Consequently,a grid model for high-accuracy coordinate transformation of CGCS2000 is proposed.Specifically,we firstly analyze a minimum curvature equation of coordinate transformation,which possesses the characteristics of both the global and local smoothness,achieving better consistency with the consecutive smoothness for the coordinate transformation of map’s linear feature.Then an iterative calculation method of grid nodes and an approach for establishing regional grid models based on collocation by two-step minimization are proposed.Meanwhile,a data structure of grid model is constructed.Finally we give the optimized grid interval and transformation accuracy in China corresponding to the proposed grid model.Using 48 433 points of 2000 National Geodetic Control Network of China,we take the proposed model into practice by constructing grid models for coordinate transformation from BJS54 and XAS80 to CGCS2000,and the external positional accuracies for both models are 0.26 m and 0.03 m respectively.

The Functional Gradient Description Method of Space Coordinate Transformation

The 3-dimension coordinate datum transformation is very important in geomatics.Generally,there are small angle coordinate transformation model (i.e.,Bursa model,et al) and large angle coordinate transformation model (i.e.,Rodrigo matrix model,et al).However,these papers only focus on the fixed and static situation regardless of the small or large angle.This paper proposes the conception of coordinate transformation gradient field,which can realize the space coordinate transformation from small angle to arbitrary angle,and from static to dynamical.Based on the equivalent characteristics of the unit quaternion rotation matrix and the Rodrigo matrix,through studying the mathematical relationship between the spatial coordinate transformation and the functional gradient,we derive a functional gradient expression of the arbitrary transformation formula in space.The study shows that the essence of spatial coordinate transformation is a kind of potential field mathematically.This potential field conception can unify all the space coordinate transformations.It is the theoretical foundation for the further study of time continuous space coordinate transformation,and this study gives a new solution for the attitude determination of motion carriers.

A VERTICAL COORDINATE TRANSFORMATION FOR 3-D NUMERICAL MODELLING OF OCEAN CIRCULATION

A so called sine-sigma vertical coordinate transformation is proposed. A vertical implicit scheme is de-rived by using cubic-spline representation of vertical current profiles. The derived 3-D nurmericalmodel has been applied to simulate the water leved field and the vertical structures of wind-induced cur-rents in an enclosed rectangular region, and the tides in the Bohai Sea. The model is believed to besatisfactory after comparing its results with an analytical soclution and observed data from tidal staions.

Evaluation of the Direct UTM Coordinates Transformation Method Based on the Standard 7-Parameters Transformation

GNSS(global navigation satellite system)observations produce the geodetic position including latitude,longitude,and altitude(or ellipsoidal height)concerning the global reference datum WGS84(Word Geodetic System 1984),which usually should be converted to another local datum to get the desired position meaning in a physical sense,coordinates of points in the local datum are usually calculated by the seven-parameter transformation method.This paper aims to validate the methods of position transformation between WGS84 and the Iraqi local datum Karbala 1979 using the UTM(universal transverse Mercator)projected coordinates directly.The proposed algorithm was tested for 10 ground control points in Erbil city and many selected points in other different cities over all Iraqi territory.The control points are measured by the CHCNAV i73 GNSS receiver.For the evaluation procedure,the RMSE(root mean square error)of the transformed coordinates is calculated with an average value of±10.715 m as an estimated uncertainty of the direct UTM coordinates transformation method over Erbil city territory,and more than±12 m over different places over Iraqi territory.

A Modified T/2 Fractionally Spaced Coordinate Transformation Blind Equalization Algorithm

When T/2 Fractionally Spaced blind Equalization Algorithm based Constant Modulus Algorithm (T/2-FSE- CMA) is employed for equalizing higher order Quadrature Amplitude Modulation signals (QAM), it has disadvantages of low convergence speed and large Mean Square Error (MSE). For overcoming these disadvantages, a Modified T/2 Fractionally Spaced blind Equalization algorithm based on Coordinate Transformation and CMA (T/2-FSE-MCTCMA) was proposed by analyzing the character of 16QAM signal constellations. In the proposed algorithm, real and imaginary parts of input signal of T/2 fractionally spaced blind equalizer are equalized, respectively, and output signals of equalizer are transformed to the same unit circle by coordinate transformation method, a new error function is defined after making coordinate transformation and used to adjust weight vector of T/2 fractionally spaced blind equalizer. The proposed algorithm can overcome large misjudgments of T/2 fractionally spaced blind equalization algorithm for equalizing multi-modulus higher order QAM. Simulation results with underwater acoustic channel models demonstrate that the proposed T/2-FSE-MCTCMA algorithm outperforms T/2 Fractionally Spaced blind Equalization algorithm bas-ed on Coordinate Transformation and CMA (T/2-FSE-CTCMA) and the T/2-FSE-CMA in convergence rate and MSE.

SARC Model for Three-Dimensional Coordinate Transformation

In this paper, a transformation model named SARC(static-filter adjustment with restricted condition) is presented, which is more practical and more rigorous in theory and fitting any angle of rotation parameter. The transformation procedure is divided into 4 steps: ① the original and object coordinates can be regarded as observations with errors; ② rigorous formula is firstly deduced in order to compute the first approximation of the transformation parameters by use of four common points and the transformation equation is linearized; ③ calculate the most probable values and variances of the seven transformation parameters by SARC model; ④ to demonstrate validity of SARC , an example is given.

Coordinate Transformation and Exact Solutions of Schrodinger Equation with Position-Dependent Effective Mass

Using the coordinate transformation method, we solve the one-dimensional Schrodinger equation with position-dependent mass. The explicit expressions for the potentials, energy eigenvalues, and eigenfunctions of the systems are given. The eigenfunctions can be expressed in terms of the Jacobi, Hermite, and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term Vm, which is produced from the dependence of mass on the position, compared with those for the systems of constant mass. The properties of Vm for several mass functions are discussed.

A New Regularized Solution to Ill-Posed Problem in Coordinate Transformation

Coordinates transformation is generally required in GPS applications. If the transformation parameters are solved with the known coordinates in a small area using the Bursa model, the precision of transformed coordinates is generally very poor. Since the translation parameters and rotation parameters are highly correlated in this case, a very large condition number of the coefficient matrix A exists in the linear system of equations. Regularization is required to reduce the effects caused by the intrinsic ill-conditioning of the problem and noises in the data, and to stabilize the solution. Based on advanced regularized methods, we propose a new regularized solution to the ill-posed coordinate transformation problem. Simulation numerical experiments of coordinate transformation are given to shed light on the relationship among different regularization approaches. The results indicate that the proposed new method can obtain a more reasonable resolution with higher precision and/or robustness.

Uzbekistan's coordinate system transformation from CS42 to WGS84 using distortion grid model

Currently,a non-geocentric geodetic coordinate system introduced in 1942(CS42),based on the Krasovsky ellipsoid and the Baltic system of normal heights introduced in 1977,is used for geodetic works in the Republic of Uzbekistan.The development of the GNSS network proposes a task of transition to a new geocentric system based on the World Geodetic System 1984(WGS84)ellipsoid.Many software products adopt 3-or 7-parameter conversions,which can cause errors up to several meters and are not suitable for many applications.In this case,local transformations using a grid of differences between observation points with known coordinates in both datums would give the best accuracy.In this paper,we discuss various interpolation methods(Kriging,Minimum Curvature,Inverse Distance to a Power and Radial Basis Function)to solve the distortion modeling between CS42 and WGS84 systems for national datum improvement.The results show that the distortion models share a common tendency for all interpolation methods:the maximum horizontal displacements are concentrated along the West Tien Shan lineament,which is the boundary of the relief lowering.The discrepancies between the grid-based(calculated)and GPS-measured coordinates are evaluated.Statistical and spatial analysis has confirmed that for the co-ordinate transformation from CS42 to WGS84 and vice versa,grid-based transformation with Radial Basis Function interpolation has a high accuracy transformation.Analysis of the available data across the eastern part of the country shows that some positional distortions existed between the CS42 and WGS84 datums.For the best RBF method,the magnitude of these distortions is about 0.019-0.755 m with a standard deviation of 0.015 m.

Water Wave Focusing Using Coordinate Transformation

In this paper, wave focusing based on a coordinate transformation is proposed. It is known that the 2-dimensional wave equation which governs a shallow water problem in a potential theory can keep invariance under coordinate transformation. Once equivalent medium parameters are obtained so as to keep the invariance, wave rays can be arbitrarily designed. We show the design of equivalent medium for wave squeezing to focus waves on a specific domain. Numerical computations are carried out by a finite element based software COMSOL Multiphysics. Results show good agreement between predictions from the theory and computations. It can be applied for a wide range of frequency because the proposed method is able to be applied regardless of the frequency.

滚仰式光电吊舱地理导引实现与误差分析

为了提升地理导引精度和成功率,根据滚仰式光电吊舱的结构特点,通过建立坐标系统、目标坐标求解和框架角解算三个步骤,完整构建了地理导引的数学模型,并在此基础上引入了速度前馈和小区域搜索模式。同时,对受惯导测量误差和目标距离影响的框架角解算误差进行了数据仿真。结果表明:经纬误差和航向角误差对俯仰角的解算误差影响较大;而高程误差和水平姿态角误差则对横滚角的解算误差影响较大。提升惯导的定位精度,可进一步减小框架角解算误差,从而提升地理导引精度,而当航向角<0.1°,水平姿态角<0.05°时,姿态角误差的影响权重会变小,再通过减小姿态角误差提升导引精度的作用已不明显。随着目标距离的增大,框架角解算误差会急剧减小。最后进行了导引试验,其俯仰、横滚角均方差均小于0.12°,表明了算法的准确性和仿真分析的有效性。

工业机械臂辅助机器视觉的工件识别与定位

传统的工业机器人系统由于缺乏感知判断能力,很难适应目标种类和位置的变化,而现代工业需要生产不同种类和规格的产品。为此,以多品种小批量生产的零件为背景提出了一种机械臂辅助机器视觉系统,该系统能够实现对工件的图像采集并进行预处理,通过改进的Canny轮廓的提取算法与基于Hausdorff距离的模版匹配完成对不同种类工件进行识别和定位。最后经实验的验证该系统的定位精度在0.35 mm内,基本满足了在实际生产中对小批量生产、不同形状且没有固定夹具的目标工件的抓取、搬运与检测等要求。