Total least squares fitting of Bézier and B-spline curves to ordered data

2002 ◽  
Vol 19 (4) ◽  
pp. 275-289 ◽  
Author(s):  
Carlos F. Borges ◽  
Tim Pastva
Keyword(s):  
Least Squares ◽  
Total Least Squares ◽  
B Spline ◽  
Least Squares Fitting ◽  
Spline Curves ◽  
Ordered Data

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.

scholarly journals Dust Removal from 3D Point Cloud Data in Mine Plane Areas Based on Orthogonal Total Least Squares Fitting and GA-TELM

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Jingli Wang ◽  
Huiyuan Zhang ◽  
Jingxiang Gao ◽  
Dong Xiao
Keyword(s):  
Least Squares ◽  
Point Cloud ◽  
Cloud Model ◽  
Hollow Structure ◽  
Total Least Squares ◽  
3D Point Cloud ◽  
Denoising Method ◽  
Least Squares Fitting ◽  
Cloud Models ◽  
Point Cloud Model

With the further development of the construction of “smart mine,” the establishment of three-dimensional (3D) point cloud models of mines has become very common. However, the truck operation caused the 3D point cloud model of the mining area to contain dust points, and the 3D point cloud model established by the Context Capture modeling software is a hollow structure. The previous point cloud denoising algorithms caused holes in the model. In view of the above problems, this paper proposes the point cloud denoising method based on orthogonal total least squares fitting and two-layer extreme learning machine improved by genetic algorithm (GA-TELM). The steps are to separate dust points and ground points by orthogonal total least squares fitting and use GA-TELM to repair holes. The advantages of the proposed method are listed as follows. First, this method could denoise without generating holes, which solves engineering problems. Second, GA-TELM has a better effect in repairing holes compared with the other methods considered in this paper. Finally, this method starts from actual problems and could be used in mining areas with the same problems. Experimental results demonstrate that it can remove dust spots in the flat area of the mine effectively and ensure the integrity of the model.

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