Rotation- and Translation-Free Estimations of Symmetric, Rank-Two Tensors with a Case Study in LIDAR Surveying

Abstract

An eigenparameter analysis plays an important role in varied fields where a symmetric tensor is involved. This technique allows one to investigate the principal behaviors (i.e., magnitudes and orientations) of a physical phenomenon that can be represented as a rank-two symmetric tensor. In this study, an analytical approach that enables rotation- and translation-free estimations of the eigenparameters from a symmetric tensor is developed, with a goal to remove the errors associated with a neglect and/or miscalculation of reference frame variations during a dynamical process. Two numerical examples, one with simulated data and the other with real light detection and ranging (LIDAR) surveying data, have been carried out to demonstrate the capability of the proposed approach in estimating the principal strains from a symmetric strain tensor. The results reveal that the proposed approach is capable of giving a direct estimate for the strain tensor without being affected by the rotation and translation of the reference frame and thus produces a principal strain estimate of a higher quality.