Least Squares Adjustment with a Rank-Deficient Weight Matrix and Its Applicability to Image/Lidar Data Processing

2021 ◽  
Vol 87 (10) ◽  
pp. 717-733 ◽  
Author(s):  
Radhika Ravi ◽  
Ayman Habib
Keyword(s):  
Least Squares ◽  
Point Cloud ◽  
Weight Matrix ◽  
Nuisance Parameters ◽  
Rank Deficiency ◽  
Naturally Occurring ◽  
Least Squares Adjustment ◽  
Line Fitting ◽  
Effective Contribution

This article proposes a solution to special least squares adjustment (LSA) models with a rank-deficient weight matrix, which are commonly encountered in geomatics. The two sources of rank deficiency in weight matrices are discussed: naturally occurring due to the inherent characteristics of LSA mathematical models and artificially induced to eliminate nuisance parameters from LSA estimation. The physical interpretation of the sources of rank deficiency is demonstrated using a case study to solve the problem of 3D line fitting, which is often encountered in geomatics but has not been addressed fully to date. Finally, some geomatics-related applications—mobile lidar system calibration, point cloud registration, and single-photo resection—are discussed along with respective experimental results, to emphasize the need to assess LSA models and their weight matrices to draw inferences regarding the effective contribution of observations. The discussion and results demonstrate the vast applications of this research in geomatics as well as other engineering domains.

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.

3D point‐cloud spatial expansion by total least‐squares line fitting

2020 ◽  
Vol 35 (172) ◽  
pp. 509-527
Author(s):  
Paweł S. Dąbrowski ◽  
Marek H. Zienkiewicz
Keyword(s):  
Least Squares ◽  
Point Cloud ◽  
Total Least Squares ◽  
3D Point Cloud ◽  
Spatial Expansion ◽  
Line Fitting

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 MATLAB Virtual Toolbox for Retrospective Rockfall Source Detection and Volume Estimation Using 3D Point Clouds: A Case Study of a Subalpine Molasse Cliff

Geosciences ◽  
2021 ◽  
Vol 11 (2) ◽  
pp. 75
Author(s):  
Dario Carrea ◽  
Antonio Abellan ◽  
Marc-Henri Derron ◽  
Neal Gauvin ◽  
Michel Jaboyedoff
Keyword(s):  
Point Cloud ◽  
Natural Hazard ◽  
Point Clouds ◽  
Volume Estimation ◽  
Natural Phenomena ◽  
Frequency Distributions ◽  
Erosion Rates ◽  
3D Point Clouds

The use of 3D point clouds to improve the understanding of natural phenomena is currently applied in natural hazard investigations, including the quantification of rockfall activity. However, 3D point cloud treatment is typically accomplished using nondedicated (and not optimal) software. To fill this gap, we present an open-source, specific rockfall package in an object-oriented toolbox developed in the MATLAB® environment. The proposed package offers a complete and semiautomatic 3D solution that spans from extraction to identification and volume estimations of rockfall sources using state-of-the-art methods and newly implemented algorithms. To illustrate the capabilities of this package, we acquired a series of high-quality point clouds in a pilot study area referred to as the La Cornalle cliff (West Switzerland), obtained robust volume estimations at different volumetric scales, and derived rockfall magnitude–frequency distributions, which assisted in the assessment of rockfall activity and long-term erosion rates. An outcome of the case study shows the influence of the volume computation on the magnitude–frequency distribution and ensuing erosion process interpretation.

Supermatrix Approach to Least-Squares Adjustment

1968 ◽  
Vol 22 (5) ◽  
pp. 22
Author(s):  
Irving H. Siegel
Keyword(s):  
Least Squares ◽  
Least Squares Adjustment

scholarly journals Evolution of halophytes: multiple origins of salt tolerance in land plants

2010 ◽  
Vol 37 (7) ◽  
pp. 604 ◽  
Author(s):  
Timothy J. Flowers ◽  
Hanaa K. Galal ◽  
Lindell Bromham
Keyword(s):  
Salt Tolerance ◽  
Novel Species ◽  
Complex Trait ◽  
Land Plants ◽  
Salt Tolerant ◽  
Physiological Mechanisms ◽  
Multiple Origins ◽  
Naturally Occurring ◽  
Tolerant Plants

The evolution of salt tolerance is interesting for several reasons. First, since salt-tolerant plants (halophytes) employ several different mechanisms to deal with salt, the evolution of salt tolerance represents a fascinating case study in the evolution of a complex trait. Second, the diversity of mechanisms employed by halophytes, based on processes common to all plants, sheds light on the way that a plant’s physiology can become adapted to deal with extreme conditions. Third, as the amount of salt-affected land increases around the globe, understanding the origins of the diversity of halophytes should provide a basis for the use of novel species in bioremediation and conservation. In this review we pose the question, how many times has salt tolerance evolved since the emergence of the land plants some 450–470 million years ago? We summarise the physiological mechanisms underlying salt-tolerance and provide an overview of the number and diversity of salt-tolerant terrestrial angiosperms (defined as plants that survive to complete their life cycle in at least 200 mM salt). We consider the evolution of halophytes using information from fossils and phylogenies. Finally, we discuss the potential for halophytes to contribute to agriculture and land management and ask why, when there are naturally occurring halophytes, it is proving to be difficult to breed salt-tolerant crops.

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