Least Squares Adjustment with a Rank-Deficient Weight Matrix and Its Applicability to Image/Lidar Data Processing

This article proposes a solution to special least squares adjustment (LSA) models with a rank-deficient weight matrix, which are commonly encountered in geomatics. The two sources of rank deficiency in weight matrices are discussed: naturally occurring due to the inherent characteristics of LSA mathematical models and artificially induced to eliminate nuisance parameters from LSA estimation. The physical interpretation of the sources of rank deficiency is demonstrated using a case study to solve the problem of 3D line fitting, which is often encountered in geomatics but has not been addressed fully to date. Finally, some geomatics-related applications—mobile lidar system calibration, point cloud registration, and single-photo resection—are discussed along with respective experimental results, to emphasize the need to assess LSA models and their weight matrices to draw inferences regarding the effective contribution of observations. The discussion and results demonstrate the vast applications of this research in geomatics as well as other engineering domains.