Total Least-Squares (TLS) for geodetic straight-line and plane adjustment

The adjustment of a straight line through a cloud of points is analyzed for the case where all coordinates are measured quantities and thus affected by random errors. The so-called Total Least-Squares Solution (TLSS) can then be obtained by solving the resulting non-linear normal equations via a newly developed iterative approximation algorithm or, equivalently, by following the smallest eigenvalue approach. After showing the superior performance of the TLS approach in the 2D-case, the procedure is extended to the 3D case where a plane is sought that best fits an observed point cloud. This procedure is implemented in surface reconstruction from a cloud of LIDAR points.