Total Least Squares Spline Approximation

Abstract

Spline approximation, using both values y i and x i as observations, is of vital importance for engineering geodesy, e.g., for approximation of profiles measured with terrestrial laser scanners, because it enables the consideration of arbitrary dispersion matrices for the observations. In the special case of equally weighted and uncorrelated observations, the resulting error vectors are orthogonal to the graph of the spline function and hence can be utilized for deformation monitoring purposes. Based on a functional model that uses cubic polynomials and constraints for continuity, smoothness and continuous curvature, the case of spline approximation with both the values y i and x i as observations is considered. In this case, some of the columns of the functional matrix contain observations and are thus subject to random errors. In the literature on mathematics and statistics this case is known as an errors-in-variables (EIV) model for which a so-called “total least squares” (TLS) solution can be computed. If weights for the observations and additional constraints for the unknowns are introduced, a “constrained weighted total least squares” (CWTLS) problem is obtained. In this contribution, it is shown that the solution for this problem can be obtained from a rigorous solution of an iteratively linearized Gauss-Helmert (GH) model. The advantage of this model is that it does not impose any restrictions on the form of the functional relationship between the involved quantities. Furthermore, dispersion matrices can be introduced without limitations, even the consideration of singular ones is possible. Therefore, the iteratively linearized GH model can be regarded as a generalized approach for solving CWTLS problems. Using a numerical example it is demonstrated how the GH model can be applied to obtain a spline approximation with orthogonal error vectors. The error vectors are compared with those derived from two least squares (LS) approaches.

Joint Stochastic Spline and Autoregressive Identification Aiming Order Reduction Based on Noisy Sensor Data

“…In the first case, the spline function is a data interpolator and can be determined by algebraic methods. In the second case, the spline function is a data approximation model or a curve fitting model and can be determined by variational methods [ 10 , 11 ]. Usually, the curve fitting is realized by means of Least Squares ( LS ) method [ 7 , 12 , 13 , 14 ].…”

“…In [ 22 ], the identification of impulse or frequency response by using modified B-splines is investigated, but the choice of the basis functions to build the spline curve is not clearly specified and the case study rather is based on toy examples. In [ 11 ], a LS adjustment is adopted for spline approximation and the knots are equidistantly rearranged by an iterative process until some quality criterion is met. Still, the data selection cannot be considered adaptive.…”

Orthogonale Regression – Realität oder Isotropie?

“…Die Verbreitung der orthogonalen Regression liegt vornehmlich in ihrem hohen Anschauungsgrad [15]. Wijewickrema et al [19] bezeichnen die Minimierung der Normalenabstände als ein natürliches Bewertungsmaß, da den geschätzten Modellparametern eine physische oder geometrische Bedeutung zugeordnet werden kann [6].…”

How to account for temporal correlations with a diagonal correlation model in a nonlinear functional model: a plane fitting with simulated and real TLS measurements

“…Terrestrial laser scanners (TLS) capture a large number of three-dimensional (3D) points rapidly, with high precision and spatial resolution. The point clouds obtained can be visualized in specific commercial software or approximated with parametric models, among which are the increasingly popular and flexible B-spline curves or surfaces (e.g., Bureick et al 2016;Koch 2010;Neitzel et al 2019). The main advantage of point cloud modelization strategies over visualization comes from the possibility of performing rigorous test statistics, for example, for deformation analysis purposes (Zhao et al 2019;Kermarrec et al 2020a).…”