Weighted total least squares for rigid body transformation and comparative study on heteroscedastic points

2011 ◽  
Author(s):  
Yongjun Zhou ◽  
Caihua Deng ◽  
Jianjun Zhu
Keyword(s):  
Comparative Study ◽  
Rigid Body ◽  
Least Squares ◽  
Total Least Squares ◽  
Weighted Total Least Squares ◽  
Rigid Body Transformation

Total Least-Squares Determination of Body Segment Attitude

2021 ◽  
Author(s):  
John Challis
Keyword(s):  
Rigid Body ◽  
Least Squares ◽  
Total Least Squares ◽  
Body Segment ◽  
Noise Levels ◽  
Least Squares Estimate ◽  
Body Attitude ◽  
Transformation Parameters ◽  
Marker Cluster ◽  
Rigid Body Transformation

Abstract To examine segment and joint attitudes when using image based motion capture it is necessary to determine the rigid body transformation parameters from an inertial reference frame to a reference frame fixed in a body segment. Determine the rigid body transformation parameters must account for errors in the coordinates measured in both reference frames, a total least-squares problem. This study presents a new derivation that shows that a singular value decomposition based method provides a total least-squares estimate of rigid body transformation parameters. The total least-squares method was compared with an algebraic method for determining rigid body attitude (TRIAD method). Two cases were examined: Case 1 where the positions of a marker cluster contained noise after the transformation, and Case 2 where the positions of a marker cluster contained noise both before and after the transformation. The white noise added to position data had a standard deviation from zero to 0.002 m, with 101 noise levels examined. For each noise level 10000 criterion attitude matrices were generated. Errors in estimating rigid body attitude were quantified by computing the angle, error angle, required to align the estimated rigid body attitude with the actual rigid body attitude. For both methods and cases as the noise level increased the error angle increased, with errors larger for Case 2 compared with Case 1. The SVD based method was superior to the TRIAD algorithm for all noise levels and both cases, and provided a total least-squares estimate of body attitude.

Fitting Weighted Total Least-Squares Planes and Parallel Planes to Support Tolerancing Standards

2012 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Point Contact ◽  
Coordinate Measuring Machine ◽  
Total Least Squares ◽  
Point Sampling ◽  
Least Squares Fitting ◽  
Weighted Total Least Squares ◽  
Measuring Machine ◽  
Fitting In ◽  
Value Decomposition

We present elegant algorithms for fitting a plane, two parallel planes (corresponding to a slot or a slab) or many parallel planes in a total (orthogonal) least-squares sense to coordinate data that is weighted. Each of these problems is reduced to a simple 3×3 matrix eigenvalue/eigenvector problem or an equivalent singular value decomposition problem, which can be solved using reliable and readily available commercial software. These methods were numerically verified by comparing them with brute-force minimization searches. We demonstrate the need for such weighted total least-squares fitting in coordinate metrology to support new and emerging tolerancing standards, for instance, ISO 14405-1:2010. The widespread practice of unweighted fitting works well enough when point sampling is controlled and can be made uniform (e.g., using a discrete point contact Coordinate Measuring Machine). However, we demonstrate that nonuniformly sampled points (arising from many new measurement technologies) coupled with unweighted least-squares fitting can lead to erroneous results. When needed, the algorithms presented also solve the unweighted cases simply by assigning the value one to each weight. We additionally prove convergence from the discrete to continuous cases of least-squares fitting as the point sampling becomes dense.

Bias-Corrected Weighted Total Least-Squares Adjustment of Condition Equations

2015 ◽  
Vol 141 (2) ◽  
pp. 04014013 ◽  
Author(s):  
Xiaohua Tong ◽  
Yanmin Jin ◽  
Songlin Zhang ◽  
Lingyun Li ◽  
Shijie Liu
Keyword(s):  
Least Squares ◽  
Total Least Squares ◽  
Weighted Total Least Squares ◽  
Least Squares Adjustment

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.

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