Research On Application Of Weighted Total Least Squares Theory In Coordinate Transformation

Total least squares(TLS)can effectively solve the parameter estimation that both the observation vector and coefficient matrix contain errors.For the coordinate transformation model,the solution that considering two systems of coordinate errors is one of the typical applications of TLS.Based on TLS and its extended theories,the coordinate transformation problems are investigated in depth,such as representation,selection,parameter estimation.Main contents of this thesis include:(1)According to the model characteristics of the coefficient matrix containing errors in geodetic problems,the different forms of EIV model in linear and nonlinear cases are summarized.Six basic algorithms of weighted total least squares(WTLS)are systematically derived,and these solutions are proven to be mutually equivalent.The WTLS algorithms of constrained,mixed,parameter weighted and Procrustes analysis are also studied.According to the various forms and types of the coordinate transformation model,the unified model suitable for different dimensions,parameter numbers and rotation matrix forms is proposed,which provids a convenient basis for estimation.In addition,multiple hypothesis testing and information criterion are used to analyze the selection of coordinate transformation models.(2)For the parameter estimation of coordinate transformation,in addition to WTLS,constrained WTLS and Procrtues WTLS.The barycentric WTLS algorithm combines the advantages of these three methods,which is suitable for arbitrary weight matrix and has a high computational efficiency.For the ill-posed problem in coordinate transformation with small area data,the WTLS regularization algorithm can generalize or replace other WTLS ill-posed algorithms in this case,such as ridge estimation,truncated SVD,iteration by correcting characteristic value.Therefore,an improved WTLS regularization algorithm based on approximate estimation of regularization parameters and adaptive iterative step matrix is proposed.The feasibility and effectiveness of the mixed WTLS,barycentric WTLS and numerical modification method in ill-posed of coordinate transformation are also studied.For the gross errors of the observation,the characteristics of model residuals base on WTLS are analyzed,and the robust WTLS estimation is proposed which has robust initial value by L1 estimation and adaptive critical value of weight function.Since the residuals of good observations may still be affected by gross errors,Therefore,the strategy of removing the influence of outliers and the strategy of compensating outliers are proposed.These two strategies can effectively improve the robustness and the accuracy.For the random model with heteroscedastic structure,the variance component estimation(VCE)of WTLS is derived using the principle of LS-VCE.Since the estimated variance component may be negative,a nonnegative estimation based on reparameterization is proposed.The accuracy of the nonnegative estimation may be affected by the structure of random model.In order to improve the accuracy,the reliable prior information of variance components is added to the formula.(3)For the updating of coordinate transformations,in order to make full use of prior information and avoid the influence of bad prior information on parameter estimation,two adaptive adjustments of WTLS with weighted parameters based on a single factor and classification matrix are discussed.In addition,the data fusion algorithm and the adaptive algorithm based on reliable prior information almost have the same accuracy,and the proposed four fusion algorithms of parameters have higher computational efficiency in multi-group parameters fusion.For the unified transformation of multi-coordinate systems and the transformation between multi-coordinate systems,the unified transformation of three coordinate systems and hand-eye calibration of robot vision are discussed as examples,and corresponding WTLS algorithm and constrained WTLS algorithm is derived respectively.For seamless prediction of coordinate transformation,considering all common points and new points of two coordinate systems as an example to analyze,and according to the relationship between adjusted residuals and predicted values,a general prediction formula is summarized and can be applied to any models.(4)At last,the thesis shows the partial algorithms and application of the coordinate transformation with three surveying cases.The first case analyzes the differences between the barycentric WTLS algorithm and other mainstream algorithms in the calculation of coordinate frame transformation;The second case discusses the feasibility of applying the robust estimation and seamless estimation of coordinate transformation to solve the compatibility analysis and the updating problem of known-points in the high-speed rail control network respectively;The third case compares the differences between WTLS algorithm and other algorithms in the solution of hand-eye calibration.