Weighted total least-squares with constraints: a universal formula for geodetic symmetrical transformations

Abstract

Although the analytical solutions for total leastsquares with multiple linear and single quadratic constraints were developed quite recently in different geodetic publications, these methods are restricted in number and type of constraints, and currently their computational efficiency and applications are mostly unknown. In this contribution, it is shown how the weighted total least-squares (WTLS) problem with arbitrary applicable constraints can be solved based on a Newton type methodology. This iterative process with quadratic convergence is expanded upon to become a compact solution for the WTLS with or without constraints. This compact solution is then further interpreted as a universal formula for the symmetrical adjustment of the errorsin-variables model which represents affine, similarity and rigid transformations in two-and three-dimensional space. Furthermore, statistical analysis of the constrained WTLS including the first-order approximation of precision and the bias was investigated. In order to substantiate our proposed method's applicability, it was used to solve the affine, similarity and rigid transformation problem in two-and threedimensional cases, where the structure of the coefficient matrix and multiple constraints were taken into account simultaneously.

Errors-in-Variables Anisotropic Extended Orthogonal Procrustes Analysis

“…Furthermore, constraints on the unknown parameters have been often introduced in geodetic applications. Schaffrin [15] solved a TLS problem with linear constraints, whereas the TLS solution of [16] can be applied with arbitrary constraints.…”

Weighted coordinate transformation formulated by standard least-squares theory

“…The idea of formulating the general TLS (or Gauss-Helmert model) problem using the SLS, total residuals and variance factor formulation originates from the work presented by Amiri-Simkooei and Jazaeri (2012). This work has then been supported by a series of publications in which this idea has been implemented in different WTLS problems (Amiri-Simkooei, 2013;Amiri-Simkooei and Jazaeri, 2013;Amiri-Simkooei et al, 2014, 2015, 2016. Other authors have also dealt with the WTLS problem (e.g.…”

A robust weighted total least-squares solution with Lagrange multipliers

“…In 2008, weighted total least-squares (WTLS) solution has also been proposed by Schaffrin and Wieser (2008) using Lagrange multipliers (LM) for an errors-in-variables (EIV) model with fairly general variance-covariance matrices. Since then, a number of WTLS solutions have been developed by Schaffrin and Wieser (2009), Markovsky et al (2010), Neitzel (2010), Shen et al (2011), Tong et al (2011), Amiri-Simkooei and Jazaeri (2012), Mahboub (2012), Amiri-Simkooei and Jazaeri (2013), Fang (2013), , Fang (2014cFang ( , 2015, Jazaeri et al (2014). In particular, Neitzel (2010) implemented a WTLS solution by using iteratively linearised GaussHelmert (GH) model (Pope, 1972), and Mahboub (2012) improved the WTLS solution of Schaffrin and Wieser (2008).…”