Abstract
In this contribution the fitting of a straight line to 3D point data is considered, with Cartesian coordinates xi, yi, zi as observations subject to random errors. A direct solution for the case of equally weighted and uncorrelated coordinate components was already presented almost forty years ago. For more general weighting cases, iterative algorithms, e.g., by means of an iteratively linearized Gauss–Helmert (GH) model, have been proposed in the literature. In this investigation, a new direct solution for the case of pointwise weights is derived. In the terminology of total least squares (TLS), this solution is a direct weighted total least squares (WTLS) approach. For the most general weighting case, considering a full dispersion matrix of the observations that can even be singular to some extent, a new iterative solution based on the ordinary iteration method is developed. The latter is a new iterative WTLS algorithm, since no linearization of the problem by Taylor series is performed at any step. Using a numerical example it is demonstrated how the newly developed WTLS approaches can be applied for 3D straight line fitting considering different weighting cases. The solutions are compared with results from the literature and with those obtained from an iteratively linearized GH model.
Highlights
Modern geodetic instruments, such as terrestrial laser scanners, provide the user directly with (CAD), a line, curve or surface approximation with a continuous mathematical function is required.In this contribution the fitting of a spatial straight line is discussed considering the coordinate components xi, yi, zi of each point Pi as observations subject to random errors, which results in a nonlinear adjustment problem
For an analysis of the recorded data or for a representation using computer-aided design (CAD), a line, curve or surface approximation with a continuous mathematical function is required
It was shown that the problem of fitting a straight line to 3D point data can be transformed into an eigenvalue problem
Summary
Modern geodetic instruments, such as terrestrial laser scanners, provide the user directly with. It is to clarify that the terms TLS and WTLS refer to algorithmic approaches for obtaining a least squares solution, which is either direct or iterative but without linearizing the problem by Taylor series at any step For case (ii) an iterative solution without linearizing the problem by Taylor series is derived, i.e., an iterative WTLS solution Both solutions are based on the work of Malissiovas [20], where similar algorithms have been presented for the solution of other typical geodetic tasks, such as straight line fitting to 2D point data, plane fitting to 3D point data and 2D similarity coordinate transformation. The WTLS solution for straight line fitting to 3D point data will be derived from a geodetic point of view by means of introducing residuals for all observations, formulating appropriate condition and constraint equations, setting up and solving the resulting normal equation systems
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