Error analysis of the 3D similarity coordinate transformation

The 3D similarity coordinate transformation with the Gauss–Helmert error model is investigated. The first-order error analysis of an analytical least-squares solution to this problem is developed in detail. While additive errors are assumed in the translation and scale estimates, a 3 × 1 multiplicative error vector is defined to effectively parameterize the rotation matrix estimation error. The propagation of the errors in the coordinate measurements to the errors in the estimated transformation parameters is derived step-by-step, and the formulae for calculating the variance–covariance matrix of the estimated parameters are presented.     

基于中位参数法相关观测的抗差加权整体最小二乘算法

在抗差加权整体最小二乘算法中,抗差模型的抗差性与初值的好坏关系极大,若以最小二乘或整体最小二乘估值作为初值,必定会受到粗差污染而影响其抗差性。考虑到观测向量和系数矩阵存在相关性,首先推导了部分变量误差（partial errors-in-variables,Partial EIV）模型的加权整体最小二乘算法,在此基础上提出了一种利用中位参数法求解抗差迭代初值的相关观测抗差加权整体最小二乘算法。然后采用中位参数法确定抗差初值,考虑到可能出现的粗差对观测空间与结构空间的综合影响,基于标准化残差构造权因子函数,实现其抗差解法。仿真实验结果表明,此算法具有良好的抗差性能,其参数估计结果比传统算法精度更高,且随着粗差个数的增加,其抗差稳定性较好。     

GPS随机模型最优不变二次无偏估计算法及实现

目前GPS基线解算通常采用简化的等权随机模型,在复杂观测环境下采用该模型将影响定位结果的精度和可靠性。本文结合方差-协方差分量估计理论,论述了利用最优不变二次无偏估计(BIQUE)方法建立GPS精密随机模型的迭代算法,并进行了相应的软件模块设计。算例分析结果表明,该算法用于GPS随机模型精化是可行的,与等权随机模型比较,所建立的GPS双差观测值方差-协方差阵更符合实际,能有效地提高整周模糊度的可靠性和定位精度。     

一种稳健性准则及相应的稳健估计

研究了已有的各种稳健性度量，根据稳健性的经典定义，本文建立了一种衡量稳健性的准则，根据这一准则，研究了在测量中常用的两种误差模型下的稳健结构。结果表明，在这一个准则和随机误差模型下的最优稳健估计，是李德仁教授提出的验后方差法；在这一准则与均值移动误差模型下的最优稳健估计，是具有均方误差最小的稳健估计。     

大地测量相关观测抗差估计理论

相关观测异常诊断、质量控制是测量数据处理领域亟待解决的难题之一。分别从方差膨胀模型和相关权元素压缩模型入手研究了相关观测的质量控制理论和方法；给出了误差影响函数；构造了方差膨胀函数和权因子收缩函数；利用观测量的等价协方差阵和等价权矩阵讨论了相关观测质量控制的计算方法。该等价协方差矩阵和等价权矩阵不仅保持了原有协方差矩阵和权矩阵的对称性，而且保持了原有协方差矩阵的相关性不变。计算结果表明异常观测的方差膨胀法和等价权法能有效地控制异常观测对参数估值的影响。     

Robustness and the robust estimate

The paper establishes a measure for robustness. With help of the measure the paper discusses the robust estimates under the stochastic error model and mean shift error model. The results show that under the measure and the stochastic error model the most reasonable robust estimate is the posterior variance approach proposed by Li Deren in 1983. Under the measure and the mean shift error model, the most robust estimate is the one with minimum mean squared error proposed by Zhu Jiangjun in 1991.     

Generalyzed variance-covariance propagation law formulae and application to explicit least-squares adjustments

Standard formulae overlook the contribution of a number of terms in the derivation of variance-covariance matrices for parameters in nonlinear least squares adjustment. In a large class of nonlinear mathematical models, these terms can contribute to an important error in the estimation of parameter variances. Improved formulae are derived. A numerical example is given and the use of our improved formula in the case of least-squares adjustment in the explicit case (L=F(X)) is fully documented.     

A robust linear least-squares estimation of camera exterior orientation using multiple geometric features

For photogrammetric applications, solutions to camera exterior orientation problem can be classified into linear (direct) and non-linear. Direct solutions are important because of their computational efficiency. Existing linear solutions suffer from lack of robustness and accuracy partially due to the fact that the majority of the methods utilize only one type of geometric entity and their frameworks do not allow simultaneous use of different types of features. Furthermore, the orthonormality constraints are weakly enforced or not enforced at all. We have developed a new analytic linear least-squares framework for determining camera exterior orientation from the simultaneous use of multiple types of geometric features. The technique utilizes 2D/3D correspondences between points, lines, and ellipse–circle pairs. The redundancy provided by different geometric features improves the robustness and accuracy of the least-squares solution. A novel way of approximately imposing orthonormality constraints on the sought rotation matrix within the linear framework is presented. Results from experimental evaluation of the new technique using both synthetic data and real images reveal its improved robustness and accuracy over existing direct methods.     

总体最小二乘回归预测模型的方差分量估计

回归预测模型是对传统回归模型的进一步扩展,不仅涉及回归模型的固定参数估计,而且将模型预测纳入平差的部分内容,更加符合实际解算需求.针对在回归模型预测中经常出现待预测非公共点(自变量)含有观测误差和随机模型不准确的问题,基于EIV (errors-in-variables)模型提出了一种同时顾及所有变量观测误差的整体解法...     

On the sensitivity of the results of least-squares adjustments concerning the stochastic model

A proper perturbation theory of a mathematical model and the quantities derived by means of least-squares adjustments is indispensable if the results have to be interpreted in a wider context. The sensitivity of some characteristic results of least-squares adjustments such as the estimated values of the parameters and their variance–covariance matrix due to imminent uncertainties of the stochastic model is discussed in detail. Linearizations are used with rigorous error measures and interval mathematics. Numerical examples conclude the investigations. Received: 27 December 1997 / Accepted: 19 April 1999     

一种地面激光雷达点云与纹理影像稳健配准方法

为了使地面激光雷达点云得到真实和高分辨率纹理,需要对目标进行全方位的拍照。这时点云和影像的配准会出现大角度的问题。同时,由于控制点选点误差、仪器误差等,配准精度往往不能满足要求。针对以上问题,本文提出一种稳健的配准方法。首先,应用重心化的空间相似变换模型和正交旋转矩阵与反对称矩阵的关系推导解算配准参数角元素的模型。将配准结果当作初始值。然后,应用一种基于共线方程的改进丹麦法选权迭代法进行降权精确配准,确定配准参数的精确值。试验表明,本文方法稳定性强、配准精度高,适合任意角度影像与点云的高精度配准。     

线阵卫星遥感影像外方位元素对偶四元数求解法

基于对偶四元数可统一描述位置与姿态的特点,提出了利用对偶四元数求解线阵卫星遥感影像外方位元素的方法。该方法使用对偶四元数的实部描述传感器的姿态,并利用对偶部和实部共同描述成像传感器的位置。通过对位置和姿态的内插建立了基于对偶四元数的外方位元素模型。为减少运算,将球面线性插值进行化简,进而建立了基于线性插值的成像几何模型。为求解外方位元素,首先对共线条件方程进行了线性化,然后通过矩阵微分运算推导了线元素的虚拟观测方程,并根据误差传播定律确定其权值,最终采用具有约束条件的参数平差法求解外方位元素。试验结果表明本文方法正确可靠,与采用欧拉角和单位四元数的外方位元素求解方法相比,有更高的参数解算精度,同时也表明了准确的初值和虚拟观测方程对外方位元素求解的必要性。     

Robust estimator for correlated observations based on bifactor equivalent weights

 A new robust parameter estimator for the adjustment of correlated observations is developed based on a `bifactor reduction' model of weight elements. A shrinking factor for weight elements is proposed. The new equivalent weight matrix composed by the bifactor weight elements preserves the symmetry and keeps the original correlation coefficients unchanged. The new parameter estimator with its error influence function is derived. The robustness and efficiency of the new robust estimator is demonstrated with a simulated example and some conclusions are drawn. Received: 5 March 2001 / Accepted: 17 January 2002     

A dual quaternion-based,closed-form pairwise registration algorithm for point clouds

The representation of similarity transformation in three-dimensional (3D) space, especially of orientation, is a crucial issue in navigation, geodesy, photogrammetry, robot arm manipulation, etc. Considering the large amount of computer resources required by iterative algorithms designed for spatial similarity transformation, the high dependence on initial values of unknown parameters, and the instability of solving transformation parameters for large-angle registration, a closed-form solution for pairwise light detection and ranging (LiDAR) point cloud registration is proposed. In this solution, dual-number quaternions are used to represent the 3D rotation. The relationship between the rotation matrix-based representation of similarity transformation and the dual quaternion-based representation is described first. Considering that the same features from two neighboring stations coincide after pairwise registration, a dual quaternion-based error norm, which is associated with the sum of the position errors, is constructed. Based on theory of least squares and by extreme value analysis of the error norm, detailed derivations of the model and the main formulas are obtained. Once the similarities between the same features from the two neighboring LiDAR stations are constructed, the rotation matrix, the scale parameter, and the translation vector are simultaneously derived. Two experiments are conducted to verify the feasibility and effectiveness of the proposed algorithm. The proposed algorithm has the advantages of simplicity and ease of implementation, making it better than the traditional methods that use matrices to describe spatial rotation. Moreover, it solves the transformation parameters without the initial estimates of unknown parameters, making it better than iterative algorithms. Most importantly, in contrast to unit quaternion-based algorithms, the proposed algorithm solves seven unknown parameters simultaneously. Therefore, it effectively avoids the accumulation of introduced error in calculation and the negative impact from the inappropriate choice of initial values.     

Efficient gravity field recovery from GOCE gravity gradient observations

 An efficient algorithm is proposed for gravity field recovery from Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) satellite gravity gradient observations. The mathematical model is formulated in the time domain, which allows the inclusion of realistic observational noise models. The algorithm combines the iterative solution of the normal equations, using a Richardson-type iteration scheme, with the fast computation of the right-hand side of the normal equations in each iteration step by a suitable approximation of the design matrix. The convergence of the iteration is investigated, error estimates are provided, and the unbiasedness of the method is proved. It is also shown that the method does not converge to the solution of the normal equations. The performance of the approach for white noise and coloured noise is demonstrated along a simulated GOCE orbit up to spherical harmonic degree and order 180. The results also indicate that the approximation error may be neglected. Received: 30 November 1999 / Accepted: 31 May 2000     

Stochastic assessment of GPS carrier phase measurements for precise static relative positioning

 Global positioning system (GPS) carrier phase measurements are used in all precise static relative positioning applications. The GPS carrier phase measurements are generally processed using the least-squares method, for which both functional and stochastic models need to be carefully defined. Whilst the functional model for precise GPS positioning is well documented in the literature, realistic stochastic modelling for the GPS carrier phase measurements is still both a controversial topic and a difficult task to accomplish in practice. The common practice of assuming that the raw GPS measurements are statistically independent in space and time, and have the same accuracy, is certainly not realistic. Any mis-specification in the stochastic model will inevitably lead to unreliable positioning results. A stochastic assessment procedure has been developed to take into account the heteroscedastic, space- and time-correlated error structure of the GPS measurements. Test results indicate that the reliability of the estimated positioning results is improved by applying the developed stochastic assessment procedure. In addition, the quality of ambiguity resolution can be more realistically evaluated. Received: 13 February 2001 / Accepted: 3 September 2001     

On symmetrical three-dimensional datum conversion

A 3-D similarity transformation is frequently used to convert GPS-WGS84-based coordinates to those in a local datum using a set of control points with coordinate values in both systems. In this application, the Gauss-Markov (GM) model is often employed to represent the problem, and a least-squares approach is used to compute the parameters within the mathematical model. However, the Gauss–Markov model considers the source coordinates in the data matrix (A) as fixed or error-free; this is an imprecise assumption since these coordinates are also measured quantities and include random errors. The errors-in-variables (EIV) model assumes that all the variables in the mathematical model are contaminated by random errors. This model may be solved using the relatively new total least-squares (TLS) estimation technique, introduced in 1980 by Golub and Van Loan. In this paper, the similarity transformation problem is analyzed with respect to the EIV model, and a novel algorithm is described to obtain the transformation parameters. It is proved that even with the EIV model, a closed form Procrustes approach can be employed to obtain the rotation matrix and translation parameters. The transformation scale may be calculated by solving the proper quadratic equation. A numerical example and a practical case study are presented to test this new algorithm and compare the EIV and the GM models.     

A priori fully populated covariance matrices in least-squares adjustment—case study: GPS relative positioning

In this contribution, using the example of the Mátern covariance matrices, we study systematically the effect of apriori fully populated variance covariance matrices (VCM) in the Gauss–Markov model, by varying both the smoothness and the correlation length of the covariance function. Based on simulations where we consider a GPS relative positioning scenario with double differences, the true VCM is exactly known. Thus, an accurate study of parameters deviations with respect to the correlation structure is possible. By means of the mean-square error difference of the estimates obtained with the correct and the assumed VCM, the loss of efficiency when the correlation structure is missspecified is considered. The bias of the variance of unit weight is moreover analysed. By acting independently on the correlation length, the smoothness, the batch length, the noise level, or the design matrix, simulations allow to draw conclusions on the influence of these different factors on the least-squares results. Thanks to an adapted version of the Kermarrec–Schön model, fully populated VCM for GPS phase observations are computed where different correlation factors are resumed in a global covariance model with an elevation dependent weighting. Based on the data of the EPN network, two studies for different baseline lengths validate the conclusions of the simulations on the influence of the Mátern covariance parameters. A precise insight into the impact of apriori correlation structures when the VCM is entirely unknown highlights that both the correlation length and the smoothness defined in the Mátern model are important to get a lower loss of efficiency as well as a better estimation of the variance of unit weight. Consecutively, correlations, if present, should not be neglected for accurate test statistics. Therefore, a proposal is made to determine a mean value of the correlation structure based on a rough estimation of the Mátern parameters via maximum likelihood estimation for some chosen time series of observations. Variations around these mean values show to have little impact on the least-squares results. At the estimates level, the effect of varying the parameters of the fully populated VCM around these approximated values was confirmed to be nearly negligible (i.e. a mm level for strong correlations and a submm level otherwise).     

等价条件平差模型的方差-协方差分量最小二乘估计方法

提出等价条件闭合差的方差-协方差分量最小二乘估计方法,简称LSV-ECM法。首先,利用等价条件平差模型建立了基于等价条件闭合差二次型的方差-协方差分量估计方程,由矩阵半拉直算子将其变换为线性Gauss-Markov形式,进而通过最小二乘准则导出了具有模型通用性、形式简洁性且满足无偏性和最优性的方差-协方差分量估计公式。其次,证明了LSV-ECM方法与残差型VCE方法的等价性,并在此基础上通过计算复杂度定量分析了所提方法的计算高效性。最后,通过边角网平差和中国区域GNSS站坐标时序建模及其结果分析,验证了所提新方法的正确性和计算高效性。     

航空重力向下延拓的最小二乘配置Tikhonov正则化法

基于最小二乘配置法向下延拓航空重力的过程中,由于协方差矩阵严重病态,影响延拓结果的稳定性和精度。针对这一问题,提出了航空重力向下延拓的最小二乘配置Tikhonov正则化法。基于全球协方差函数模型建立航空重力数据与地面重力数据的协方差关系,引入基于广义交叉验证法,选择正则化参数的Tikhonov正则化法改善协方差矩阵的病态性,抑制观测噪声对延拓结果的放大影响。基于EGM2008重力场模型,设计了山区、丘陵和海域3种不同地形区域的航空重力数据向下延拓的仿真实验,实验结果验证了该方法的有效性。