Least Squares Adjustment with Finite Residuals for Non-Linear Constraints and Partially Correlated Data

1973 ◽  
Author(s):  
Aivars Celmins
Keyword(s):  
Least Squares ◽  
Linear Constraints ◽  
Correlated Data ◽  
Non Linear ◽  
Least Squares Adjustment

Clarification of discussion on constrained refinement

1978 ◽  
Vol 34 (6) ◽  
pp. 1020-1021
Author(s):  
P. F. Price
Keyword(s):  
Least Squares ◽  
Linear Constraints ◽  
Linear Case ◽  
Linear Least Squares ◽  
Acta Cryst ◽  
Non Linear

Pawley's [Acta Cryst. (1976). A32, 921-922] disproof of the C & D method of applying constraints in crystallographic refinements [Chesick & Davidon (1975). Acta Cryst. A31, 586-591] is incorrect. Nevertheless, the C & D method is incorrect. Both the method and its disproof clearly fail for the particular case of linear least-squares with linear constraints. The method can be corrected in this case but not for the non-linear case.

scholarly journals Using Least-Squares Residuals to Assess the Stochasticity of Measurements—Example: Terrestrial Laser Scanner and Surface Modeling

2021 ◽  
Vol 5 (1) ◽  
pp. 59
Author(s):  
Gaël Kermarrec ◽  
Niklas Schild ◽  
Jan Hartmann
Keyword(s):  
Least Squares ◽  
Laser Scanner ◽  
Real Data ◽  
Point Clouds ◽  
Statistical Testing ◽  
Normal Vector ◽  
Engineering Structures ◽  
3D Point Clouds ◽  
Least Squares Adjustment ◽  
Correlation Parameters

Terrestrial laser scanners (TLS) capture a large number of 3D points rapidly, with high precision and spatial resolution. These scanners are used for applications as diverse as modeling architectural or engineering structures, but also high-resolution mapping of terrain. The noise of the observations cannot be assumed to be strictly corresponding to white noise: besides being heteroscedastic, correlations between observations are likely to appear due to the high scanning rate. Unfortunately, if the variance can sometimes be modeled based on physical or empirical considerations, the latter are more often neglected. Trustworthy knowledge is, however, mandatory to avoid the overestimation of the precision of the point cloud and, potentially, the non-detection of deformation between scans recorded at different epochs using statistical testing strategies. The TLS point clouds can be approximated with parametric surfaces, such as planes, using the Gauss–Helmert model, or the newly introduced T-splines surfaces. In both cases, the goal is to minimize the squared distance between the observations and the approximated surfaces in order to estimate parameters, such as normal vector or control points. In this contribution, we will show how the residuals of the surface approximation can be used to derive the correlation structure of the noise of the observations. We will estimate the correlation parameters using the Whittle maximum likelihood and use comparable simulations and real data to validate our methodology. Using the least-squares adjustment as a “filter of the geometry” paves the way for the determination of a correlation model for many sensors recording 3D point clouds.

scholarly journals Adaptive Robust Kernels for Non-Linear Least Squares Problems

2021 ◽  
pp. 1-1
Author(s):  
Nived Chebrolu ◽  
Thomas Labe ◽  
Olga Vysotska ◽  
Jens Behley ◽  
Cyrill Stachniss
Keyword(s):  
Least Squares ◽  
Linear Least Squares ◽  
Least Squares Problems ◽  
Non Linear ◽  
Linear Least Squares Problems
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