An α-moving total least squares fitting method for measurement data

2020 ◽  
Vol 235 (1-2) ◽  
pp. 65-72
Author(s):  
Tianqi Gu ◽  
Chenjie Hu ◽  
Dawei Tang ◽  
Shuwen Lin ◽  
Tianzhi Luo
Keyword(s):  
Least Squares ◽  
Geometric Characteristic ◽  
Measurement Data ◽  
Characteristic Parameter ◽  
Negative Influence ◽  
Total Least Squares ◽  
Least Square ◽  
Random Errors ◽  
Improved Method ◽  
Fitting Accuracy

The Moving Least Squares (MLS) and Moving Total Least Squares (MTLS) method are widely used for approximating discrete data in many areas such as surface reconstruction. One of the disadvantages of MLS is that it only considers the random errors in the dependent variables. The MTLS method achieves a better fitting accuracy by taking into account the errors of both dependent and independent variables. However, both MLS and MTLS suffer from a low fitting accuracy when applied to the measurement data with outliers. In this work, an improved method named as α-MTLS method is proposed, which uses the Total Least Square (TLS) method based on singular value decomposition (SVD) to fit the nodes in the influence domain and introduces a geometric characteristic parameter α to associated with the abnormal degree of nodes. The generated fitting points are used to construct the parameter and quantify the abnormal degree of the nodes. The node with the largest parameter value is eliminated and the remaining nodes are used to determine the local coefficients. By trimming only one node per influence domain, multiple outliers of measurement data can be effectively handled. There is no need to set threshold values subjectively or assign weights which avoids the negative influence of manual operation. The performance of the improved method is demonstrated by numerical simulations and measurement experiment. It is shown that the α-MTLS method can effectively reduce the influence of the outliers and thus has higher fitting accuracy and greater robustness than that of the MLS and MTLS method.

Comparing of Coordinate Transformation Based on Total Least Squares and Least Squares

2014 ◽  
Vol 522-524 ◽  
pp. 1211-1214
Author(s):  
Qing Wu Meng ◽  
Lu Meng
Keyword(s):  
Least Squares ◽  
Coordinate Transformation ◽  
Least Square Method ◽  
Coefficient Matrix ◽  
Total Least Squares ◽  
Least Square ◽  
Experimental Results ◽  
Transformation Models ◽  
Total Least Square

The coordinate transformation models based on least square method and total least square are built and discussed. The least square model only includes the errors of observation vectors, the total least square model simultaneously takes into consideration to the errors of observation vectors and the errors of coefficient matrix. The both models are verified and compared in experiment. The experimental results showed that the model of total least square is more in line with actual, and more reasonable than by least square theoretically, and the coordinate transformation solution result of total least square with least square is more near.

scholarly journals Weighted Total Least Squares (WTLS) Solutions for Straight Line Fitting to 3D Point Data

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1450
Author(s):  
Georgios Malissiovas ◽  
Frank Neitzel ◽  
Sven Weisbrich ◽  
Svetozar Petrovic
Keyword(s):  
Least Squares ◽  
Iterative Solution ◽  
Total Least Squares ◽  
Dispersion Matrix ◽  
Random Errors ◽  
Direct Solution ◽  
Weighted Total Least Squares ◽  
Straight Line ◽  
Point Data ◽  
Line Fitting

In this contribution the fitting of a straight line to 3D point data is considered, with Cartesian coordinates xi, yi, zi as observations subject to random errors. A direct solution for the case of equally weighted and uncorrelated coordinate components was already presented almost forty years ago. For more general weighting cases, iterative algorithms, e.g., by means of an iteratively linearized Gauss–Helmert (GH) model, have been proposed in the literature. In this investigation, a new direct solution for the case of pointwise weights is derived. In the terminology of total least squares (TLS), this solution is a direct weighted total least squares (WTLS) approach. For the most general weighting case, considering a full dispersion matrix of the observations that can even be singular to some extent, a new iterative solution based on the ordinary iteration method is developed. The latter is a new iterative WTLS algorithm, since no linearization of the problem by Taylor series is performed at any step. Using a numerical example it is demonstrated how the newly developed WTLS approaches can be applied for 3D straight line fitting considering different weighting cases. The solutions are compared with results from the literature and with those obtained from an iteratively linearized GH model.

Total Least-Squares Methods for Active View Registration of Three-Dimensional Line Laser Scanning Data

2004 ◽  
Vol 127 (1) ◽  
pp. 50-56 ◽  
Author(s):  
F. Xi ◽  
D. Nancoo ◽  
G. Knopf
Keyword(s):  
Least Squares ◽  
Laser Scanning ◽  
Least Squares Method ◽  
Three Dimensional ◽  
Measurement Data ◽  
Ordinary Least Squares ◽  
Total Least Squares ◽  
Position Measurement ◽  
Point Position ◽  
Ordinary Least Squares Method

In this paper a method is proposed to register three-dimensional line laser scanning data acquired in two different viewpoints. The proposed method is based on three-point position measurement by scanning three reference balls to determine the transformation between two views. Since there are errors in laser scanning data and sphere fitting, the two sets of three-point position measurement data at two different views are both subject to errors. For this reason, total least-squares methods are applied to determine the transformation, because they take into consideration the errors both at inputs and outputs. Simulations and experiment are carried to compare three methods, namely, ordinary least-squares method, unconstrained total least-squares method, and constrained total least-squares method. It is found that the last method gives the most accurate results.

scholarly journals A Novel Reconstruction Method for Measurement Data Based on MTLS Algorithm

Sensors ◽  
2020 ◽  
Vol 20 (22) ◽  
pp. 6449
Author(s):  
Tianqi Gu ◽  
Chenjie Hu ◽  
Dawei Tang ◽  
Tianzhi Luo
Keyword(s):  
Least Squares ◽  
Measurement Data ◽  
Local Approximation ◽  
Reconstruction Method ◽  
Random Errors ◽  
Least Trimmed Squares ◽  
Approximation Accuracy ◽  
Robust Model ◽  
Reconstruction Methods ◽  
Measurement Experiment

Reconstruction methods for discrete data, such as the Moving Least Squares (MLS) and Moving Total Least Squares (MTLS), have made a great many achievements with the progress of modern industrial technology. Although the MLS and MTLS have good approximation accuracy, neither of these two approaches are robust model reconstruction methods and the outliers in the data cannot be processed effectively as the construction principle results in distorted local approximation. This paper proposes an improved method that is called the Moving Total Least Trimmed Squares (MTLTS) to achieve more accurate and robust estimations. By applying the Total Least Trimmed Squares (TLTS) method to the orthogonal construction way in the proposed MTLTS, the outliers as well as the random errors of all variables that exist in the measurement data can be effectively suppressed. The results of the numerical simulation and measurement experiment show that the proposed algorithm is superior to the MTLS and MLS method from the perspective of robustness and accuracy.

Comparison of Total Least Squares and Least Squares for Four- and Seven-parameter Model Coordinate Transformation

2016 ◽  
Vol 10 (4) ◽  
Author(s):  
You Wu ◽  
Jun Liu ◽  
Hui Yong Ge
Keyword(s):  
Least Squares ◽  
Coordinate Transformation ◽  
Relative Effectiveness ◽  
Satellite System ◽  
Total Least Squares ◽  
Design Matrix ◽  
Errors In Variables ◽  
Random Errors ◽  
Global Navigation Satellite ◽  
Observation Vector

AbstractTotal least squares (TLS) is a technique that solves the traditional least squares (LS) problem for an errors-in-variables (EIV) model, in which both the observation vector and the design matrix are contaminated by random errors. Four- and seven-parameter models of coordinate transformation are typical EIV model. To determine which one of TLS and LS is more effective, taking the four- and seven-parameter models of Global Navigation Satellite System (GNSS) coordinate transformation with different coincidence pointsas examples, the relative effectiveness of the two methods was compared through simulation experiments. The results showed that in the EIV model, the errors-in-variables-only (EIVO) model and the errors-in-observations-only (EIOO) model, TLS is slightly inferior to LS in the four-parameter model coordinate transformation, and TLS is equivalent to LS in the seven-parameter model coordinate transformation. Consequently, in the four- and seven-parameter model coordinate transformation, TLS has no obvious advantage over LS.

scholarly journals Total Least Squares Spline Approximation

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 462 ◽  
Author(s):  
Frank Neitzel ◽  
Nikolaj Ezhov ◽  
Svetozar Petrovic
Keyword(s):  
Least Squares ◽  
Spline Approximation ◽  
Functional Model ◽  
Total Least Squares ◽  
Deformation Monitoring ◽  
Random Errors ◽  
Laser Scanners ◽  
Engineering Geodesy ◽  
Continuous Curvature ◽  
Cubic Polynomials

Spline approximation, using both values y i and x i as observations, is of vital importance for engineering geodesy, e.g., for approximation of profiles measured with terrestrial laser scanners, because it enables the consideration of arbitrary dispersion matrices for the observations. In the special case of equally weighted and uncorrelated observations, the resulting error vectors are orthogonal to the graph of the spline function and hence can be utilized for deformation monitoring purposes. Based on a functional model that uses cubic polynomials and constraints for continuity, smoothness and continuous curvature, the case of spline approximation with both the values y i and x i as observations is considered. In this case, some of the columns of the functional matrix contain observations and are thus subject to random errors. In the literature on mathematics and statistics this case is known as an errors-in-variables (EIV) model for which a so-called “total least squares” (TLS) solution can be computed. If weights for the observations and additional constraints for the unknowns are introduced, a “constrained weighted total least squares” (CWTLS) problem is obtained. In this contribution, it is shown that the solution for this problem can be obtained from a rigorous solution of an iteratively linearized Gauss-Helmert (GH) model. The advantage of this model is that it does not impose any restrictions on the form of the functional relationship between the involved quantities. Furthermore, dispersion matrices can be introduced without limitations, even the consideration of singular ones is possible. Therefore, the iteratively linearized GH model can be regarded as a generalized approach for solving CWTLS problems. Using a numerical example it is demonstrated how the GH model can be applied to obtain a spline approximation with orthogonal error vectors. The error vectors are compared with those derived from two least squares (LS) approaches.

scholarly journals On weighted total least squares adjustment for solving the nonlinear problems

2014 ◽  
Vol 4 (1) ◽  
Author(s):  
C. Hu ◽  
Y. Chen ◽  
Y. Peng
Keyword(s):  
Data Processing ◽  
Least Squares ◽  
Least Squares Method ◽  
Nonlinear Problems ◽  
Total Least Squares ◽  
Geodetic Data ◽  
Least Square ◽  
Weight Matrix ◽  
Weighted Total Least Squares ◽  
Least Squares Adjustment

AbstractIn the classical geodetic data processing, a non- linear problem always can be converted to a linear least squares adjustment. However, the errors in Jacob matrix are often not being considered when using the least square method to estimate the optimal parameters from a system of equations. Furthermore, the identity weight matrix may not suitable for each element in Jacob matrix. The weighted total least squares method has been frequently applied in geodetic data processing for the case that the observation vector and the coefficient matrix are perturbed by random errors, which are zero mean and statistically in- dependent with inequality variance. In this contribution, we suggested an approach that employ the weighted total least squares to solve the nonlinear problems and to mitigate the affection of noise in Jacob matrix. The weight matrix of the vector from Jacob matrix is derived by the law of nonlinear error propagation. Two numerical examples, one is the triangulation adjustment and another is a simulation experiment, are given at last to validate the feasibility of the developed method.

Nonlinear Measurement Data Processing Based on Improved Damping Least Squares

2011 ◽  
Vol 105-107 ◽  
pp. 2034-2038
Author(s):  
Gui Ling Li
Keyword(s):  
Data Processing ◽  
Least Squares ◽  
Accuracy Assessment ◽  
Measurement Data ◽  
Nonlinear Least Squares ◽  
Taylor Series Expansion ◽  
Least Square ◽  
High Tech ◽  
Damping Factor ◽  
Observation Data

Datum are the key of “Digital Earth”.In measurement, dealing with nonlinear models of observation datum, we may take their approximate values at observation values by Taylor series expansion, say, taking first-order item as a linear function of classical adjustment. But requirements of observation data, processing and accuracy assessment are higher and higher with today's fast-growing of high-tech mapping and surveying. So study on nonlinear least squares adjustment has been paid more and more attention. Damping least squares, as a modified algorithm of Gauss-Newton’s algorithm, is necessary to add a damping factor to improve the nature of a coefficient matrix. But it is difficult to choose a suitable damping factor, and needs to solve a group of linear equations repeatedly. In this paper, an improved damping least square was utilized for the non-linear processing of measurement datum in order to reduce a lot of computational workload.

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