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.
Highlights
To capture the geometry of arbitrary objects as completely as possible, nowadays in engineering geodesy areal measurement methods are applied
Since the distribution of the knots plays a crucial role in spline approximation, various knot placement strategies have been proposed in the literature; see, e.g., the approaches developed by Cox et al [26], Schwetlick and Schütze [27] and Park [28]
We have presented a total least squares spline approximation generated from a rigorous evaluation of an iteratively linearized Gauss-Helmert model under consideration of additional constraints for the unknowns
Summary
To capture the geometry of arbitrary objects as completely as possible, nowadays in engineering geodesy areal (area-wise, in contrast to point-wise) measurement methods are applied. Starting from (3) these condition equations can be expressed in a general form as a so-called errors-in-variables (EIV) model (in which some of the columns of the functional matrix A contain observations and are subject to random errors EA ) with constraints. Jazaeri et al [9] have presented a new flexible algorithm for solving the weighted TLS problem within an EIV model having linear and quadratic constraints and singular dispersion matrices. Their iterative algorithm is based on an equivalent translation of the objective function (13) subject to (9) and (10) into a Lagrange target function.
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